Theorem spimfv 1687

Description: Specialization, using implicit substitution. Version of spim 1726 with a disjoint variable condition. See spimv 1799 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
spimfv.nf	|- F/ x ps
spimfv.1	|- ( x = y -> ( ph -> ps ) )

Assertion

Ref	Expression
spimfv	|- ( A. x ph -> ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem spimfv

Step	Hyp	Ref	Expression
1		spimfv.nf	. 2 |- F/ x ps
2		a9ev 1685	. . 3 |- E. x x = y
3		spimfv.1	. . 3 |- ( x = y -> ( ph -> ps ) )
4	2, 3	eximii 1590	. 2 |- E. x ( ph -> ps )
5	1, 4	19.36i 1660	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   A.wal 1341   F/wnf 1448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  chvarfv  1688