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Add buildThereExists.

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commit 5377d336ad8368426a3f6d5517d519189a01285a 1 parent 8dc2bfe
Jim Kingdon authored
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  1. +10 −0 Main/I/n/t/Intuitionistic first-order logic
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10 Main/I/n/t/Intuitionistic first-order logic
@@ -602,6 +602,16 @@ thm (ForAllBiconditional () () ((∀ x (φ ↔ ψ)) → ((∀ x φ) ↔ (∀ x
))
</jh>
+=== Adding there-exists to both sides of a biconditional ===
+The counterpart to <code>buildForAll</code> follows from <code>addThereExists</code>. The proof in [[First-order logic in terms of substitution built on equality]] is based on the relationship between <code>∀</code> and <code>∃</code> which holds in classical logic but not in intuitionistic logic.
+<jh>
+thm (buildThereExists () ((H (φ ↔ ψ))) ((∃ x φ) ↔ (∃ x ψ)) (
+ H eliminateBiconditionalReverse x addThereExists
+ H eliminateBiconditionalForward x addThereExists
+ introduceBiconditionalFromImplications
+))
+</jh>
+
== Export ==
Having proved everything in [[Interface:Intuitionistic first-order logic]], we export to it.
<jh>
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