Theorem bj-nalnaleximiOLD 31798

Description: An inference for distributing quantifiers over a double implication. The general statement that speimfw 1863 proves. (Contributed by BJ, 12-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-nalnaleximiOLD.1	⊢ (𝜒 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-nalnaleximiOLD	⊢ (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem bj-nalnaleximiOLD

Step	Hyp	Ref	Expression
1		bj-nalnaleximiOLD.1	. . 3 ⊢ (𝜒 → (𝜑 → 𝜓))
2	1	eximi 1752	. 2 ⊢ (∃𝑥𝜒 → ∃𝑥(𝜑 → 𝜓))
3		df-ex 1696	. 2 ⊢ (∃𝑥𝜒 ↔ ¬ ∀𝑥 ¬ 𝜒)
4		19.35 1794	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
5	2, 3, 4	3imtr3i 279	1 ⊢ (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑 → ∃𝑥𝜓))

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:  bj-nalnalimiOLD  31799