Theorem spcimgf 2810

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 2806	. 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 1444	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   A.wal 1346   = wceq 1348   F/wnf 1453   e. wcel 2141   F/_wnfc 2299

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  bj-nn0sucALT  14013