Theorem bj-exalimi 34741

Description: An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 34740 (using mpg 1801) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1968 proves. (Contributed by BJ, 29-Sep-2019.)

Hypothesis

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

Assertion

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

Proof of Theorem bj-exalimi

Step	Hyp	Ref	Expression
1		bj-exalimi.1	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	com12 32	. . 3 ⊢ (𝜓 → (𝜑 → 𝜒))
3	2	aleximi 1835	. 2 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜒))
4	3	com12 32	1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by: (None)