Theorem bj-exaleximi 31800

Description: An inference for distributing quantifiers over a double implication. (Almost) the general statement that speimfw 1863 proves. (Contributed by BJ, 29-Sep-2019.)

Hypothesis

Ref	Expression
bj-exaleximi.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
bj-exaleximi	⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))

Proof of Theorem bj-exaleximi

Step	Hyp	Ref	Expression
1		bj-exaleximi.1	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	com12 32	. . 3 ⊢ (𝜓 → (𝜑 → 𝜒))
3	2	aleximi 1749	. 2 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜒))
4	3	com12 32	1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  bj-exalimi  31801