Theorem spcimgf 2761

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimgf.1	|- F/_ x A
spcimgf.2	|- F/ x ps
spcimgf.3	|- ( x = A -> ( ph -> ps ) )

Assertion

Ref	Expression
spcimgf	|- ( A e. V -> ( A. x ph -> ps ) )

Proof of Theorem spcimgf

Step	Hyp	Ref	Expression
1		spcimgf.2	. . 3 |- F/ x ps
2		spcimgf.1	. . 3 |- F/_ x A
3	1, 2	spcimgft 2757	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ( A. x ph -> ps ) ) )
4		spcimgf.3	. 2 |- ( x = A -> ( ph -> ps ) )
5	3, 4	mpg 1427	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   A.wal 1329   = wceq 1331   F/wnf 1436   e. wcel 1480   F/_wnfc 2266

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  bj-nn0sucALT  13165