Theorem bj-exalimi 37607

Description: An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 37606 (using mpg 1829) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1995 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 33	. . 3 ⊢ (𝜓 → (𝜑 → 𝜒))
3	2	aleximi 1864	. 2 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜒))
4	3	com12 33	1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))

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-axdd2ALT  37611