Theorem spcgf 2817

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)

Hypotheses

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

Assertion

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

Proof of Theorem spcgf

Step	Hyp	Ref	Expression
1		spcgf.2	. . 3 |- F/ x ps
2		spcgf.1	. . 3 |- F/_ x A
3	1, 2	spcgft 2812	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. V -> ( A. x ph -> ps ) ) )
4		spcgf.3	. 2 |- ( x = A -> ( ph <-> ps ) )
5	3, 4	mpg 1449	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   = wceq 1353   F/wnf 1458   e. wcel 2146   F/_wnfc 2304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737

This theorem is referenced by:  spcgv  2822  rspc  2833  elabgt  2876  eusvnf  4447  mpofvex  6194