Theorem bj-exalims 32613

Description: Distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1878 proves. (Contributed by BJ, 29-Sep-2019.)

Hypothesis

Ref	Expression
bj-exalims.1	⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))

Assertion

Ref	Expression
bj-exalims	⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)))

Proof of Theorem bj-exalims

Step	Hyp	Ref	Expression
1		bj-exalim 32611	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))
2		bj-exalims.1	. . . 4 ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
3		eximal 1707	. . . 4 ⊢ ((∃𝑥𝜒 → 𝜒) ↔ (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
4	2, 3	sylibr 224	. . 3 ⊢ (∃𝑥𝜑 → (∃𝑥𝜒 → 𝜒))
5	4	a1i 11	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∃𝑥𝜒 → 𝜒)))
6	1, 5	syldd 72	1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  bj-exalimsi  32614