Theorem bj-nalnalimiOLD 31799

Description: An inference for distributing quantifiers over a double implication. The general statement that spimfw 1865 proves. (Contributed by BJ, 12-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
bj-nalnalimiOLD	⊢ (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑 → 𝜓))

Proof of Theorem bj-nalnalimiOLD

Step	Hyp	Ref	Expression
1		bj-nalnalimiOLD.1	. . 3 ⊢ (𝜒 → (𝜑 → 𝜓))
2	1	bj-nalnaleximiOLD 31798	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑 → ∃𝑥𝜓))
3		bj-nalnalimiOLD.2	. . 3 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
4		eximal 1698	. . 3 ⊢ ((∃𝑥𝜓 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
5	3, 4	mpbir 220	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
6	2, 5	syl6 34	1 ⊢ (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑 → 𝜓))

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)