Theorem spcgv 2867

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 2350	. 2 |- F/_ x A
2		nfv 1552	. 2 |- F/ x ps
3		spcgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	spcgf 2862	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2178

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  spcv  2874  mob2  2960  intss1  3914  dfiin2g  3974  exmidsssnc  4263  exmid1stab  4268  frirrg  4415  frind  4417  alxfr  4526  elirr  4607  en2lp  4620  tfisi  4653  mptfvex  5688  tfrcl  6473  rdgisucinc  6494  frecabex  6507  fisseneq  7057  mkvprop  7286  exmidfodomrlemr  7341  exmidfodomrlemrALT  7342  acfun  7350  exmidmotap  7408  ccfunen  7411  zfz1isolem1  11022  zfz1iso  11023  uniopn  14588  pw1nct  16142  sbthom  16167