Theorem bj-exalimsi 37282

Description: An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1998 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 37281	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)))
3		bj-exalimsi.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	2, 3	mpg 1830	1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by: (None)