Theorem bj-eximcom 37608

Description: A commuted form of exim 1866 which is sometimes posited as an axiom in instuitionistic modal logic. Forward implication of 19.35 1909. Its converse is not intuitionistic. (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 43	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
2	1	aleximi 1864	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓))
3	2	com12 33	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  bj-nfimt  37614  bj-spimnfe  37615  bj-spimenfa  37616  bj-cbvalimdlem  37620  bj-cbveximdlem  37621  bj-wnf2  37714  bj-snsetex  37968  bj-axseprep  38082