Theorem bj-exalimsi 34553

Description: An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1974 proves. (Contributed by BJ, 29-Sep-2019.)

Hypotheses

Ref	Expression
bj-exalimsi.1	⊢ (𝜑 → (𝜓 → 𝜒))
bj-exalimsi.2	⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))

Assertion

Ref	Expression
bj-exalimsi	⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))

Proof of Theorem bj-exalimsi

Step	Hyp	Ref	Expression
1		bj-exalimsi.2	. . 3 ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
2	1	bj-exalims 34552	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)))
3		bj-exalimsi.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	2, 3	mpg 1805	1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 210  df-ex 1788

This theorem is referenced by: (None)