Theorem spimfv 2235

Description: Specialization, using implicit substitution. Version of spim 2387 with a disjoint variable condition, which does not require ax-13 2372. See spimvw 2000 for a version with two disjoint variable conditions, requiring fewer axioms, and spimv 2390 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 1974	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		spimfv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1840	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
5	1, 4	19.36i 2227	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1537  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-ex 1784  df-nf 1788

This theorem is referenced by:  chvarfv  2236  cbv3v2  2237  cbv3v  2334  setrec2fun  46284