Theorem spimfw 1865

Description: Specialization, with additional weakening (compared to sp 2041) 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, 7-Aug-2017.)

Hypotheses

Ref	Expression
spimfw.1	⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
spimfw.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimfw	⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → 𝜓))

Proof of Theorem spimfw

Step	Hyp	Ref	Expression
1		spimfw.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
2	1	speimfw 1863	. 2 ⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))
3		df-ex 1696	. . 3 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
4		spimfw.1	. . . 4 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
5	4	con1i 143	. . 3 ⊢ (¬ ∀𝑥 ¬ 𝜓 → 𝜓)
6	3, 5	sylbi 206	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
7	2, 6	syl6 34	1 ⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀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:  spimw  1913