Theorem bj-eximcom 34803

Description: A commuted form of exim 1839 which is sometimes posited as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.)

Assertion

Ref	Expression
bj-eximcom	⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem bj-eximcom

Step	Hyp	Ref	Expression
1		pm2.27 42	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
2	1	aleximi 1837	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓))
3	2	com12 32	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815

This theorem depends on definitions:  df-bi 206  df-ex 1786

This theorem is referenced by:  bj-wnf2  34879  bj-snsetex  35132