Theorem spcgv 2826

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 2319	. 2 |- F/_ x A
2		nfv 1528	. 2 |- F/ x ps
3		spcgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	spcgf 2821	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2148

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741

This theorem is referenced by:  spcv  2833  mob2  2919  intss1  3861  dfiin2g  3921  exmidsssnc  4205  exmid1stab  4210  frirrg  4352  frind  4354  alxfr  4463  elirr  4542  en2lp  4555  tfisi  4588  mptfvex  5603  tfrcl  6367  rdgisucinc  6388  frecabex  6401  fisseneq  6933  mkvprop  7158  exmidfodomrlemr  7203  exmidfodomrlemrALT  7204  acfun  7208  exmidmotap  7262  ccfunen  7265  zfz1isolem1  10822  zfz1iso  10823  uniopn  13586  pw1nct  14837  sbthom  14859