Theorem speimfw 1863

Description: Specialization, with additional weakening (compared to 19.2 1879) to allow bundling of 𝑥 and 𝑦. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Dec-2017.)

Hypothesis

Ref	Expression
speimfw.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
speimfw	⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem speimfw

Step	Hyp	Ref	Expression
1		df-ex 1696	. . 3 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
2	1	biimpri 217	. 2 ⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
3		speimfw.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	3	com12 32	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜓))
5	4	aleximi 1749	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜓))
6	2, 5	syl5com 31	1 ⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀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:  spimfw  1865