Theorem spimfv 2274

Description: Specialization, using implicit substitution. Version of spim 2418 with a disjoint variable condition, which does not require ax-13 2403. See spimvw 2015 for a version with two disjoint variable conditions, requiring fewer axioms, and spimv 2421 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
spimfv.nf	⊢ Ⅎ𝑥𝜓
spimfv.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimfv	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem spimfv

Step	Hyp	Ref	Expression
1		spimfv.nf	. 2 ⊢ Ⅎ𝑥𝜓
2		ax6ev 1998	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		spimfv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1866	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
5	1, 4	19.36i 2266	1 ⊢ (∀𝑥𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  Ⅎwnf 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1809  df-nf 1813

This theorem is used by:  chvarfv  2275  cbv3v2  2276  cbv3v  2366  setrec2fun  50498