Theorem spimw 1974

Description: Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)

Hypotheses

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

Assertion

Ref	Expression
spimw	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spimw

Step	Hyp	Ref	Expression
1		ax6v 1972	. 2 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2		spimw.1	. . 3 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
3		spimw.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	spimfw 1969	. 2 ⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → 𝜓))
5	1, 4	ax-mp 5	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-6 1971

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  spnfw  1983  spimvw  1999  cbvaliw  2009  spfw  2036