Theorem spcgv 2768

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)

Hypothesis

Ref	Expression
spcgv.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   ps, x   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem spcgv

Step	Hyp	Ref	Expression
1		nfcv 2279	. 2 |- F/_ x A
2		nfv 1508	. 2 |- F/ x ps
3		spcgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	spcgf 2763	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   e. wcel 1480

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:  spcv  2774  mob2  2859  intss1  3781  dfiin2g  3841  exmidsssnc  4121  frirrg  4267  frind  4269  alxfr  4377  elirr  4451  en2lp  4464  tfisi  4496  mptfvex  5499  tfrcl  6254  rdgisucinc  6275  frecabex  6288  fisseneq  6813  mkvprop  7025  exmidfodomrlemr  7051  exmidfodomrlemrALT  7052  acfun  7056  ccfunen  7072  zfz1isolem1  10576  zfz1iso  10577  uniopn  12157  exmid1stab  13184  sbthom  13210