Theorem spcgf 2855

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 2850	. 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 1474	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1483   e. wcel 2176   F/_wnfc 2335

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by:  spcgv  2860  rspc  2871  elabgt  2914  eusvnf  4501  mpofvex  6293  gropd  15677  grstructd2dom  15678