Theorem bj-exalimi 31801

Description: An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1865 proves. (Contributed by BJ, 29-Sep-2019.)

Hypotheses

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

Assertion

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

Proof of Theorem bj-exalimi

Step	Hyp	Ref	Expression
1		bj-exalimi.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	bj-exaleximi 31800	. 2 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))
3		bj-exalimi.2	. . 3 ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
4		eximal 1698	. . 3 ⊢ ((∃𝑥𝜒 → 𝜒) ↔ (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
5	3, 4	sylibr 223	. 2 ⊢ (∃𝑥𝜑 → (∃𝑥𝜒 → 𝜒))
6	2, 5	syld 46	1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:  ¬ wn 3   → 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: (None)