Theorem vtoclga 2867

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)

Hypotheses

Ref	Expression
vtoclga.1	|- ( x = A -> ( ph <-> ps ) )
vtoclga.2	|- ( x e. B -> ph )

Assertion

Ref	Expression
vtoclga	|- ( A e. B -> ps )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem vtoclga

Step	Hyp	Ref	Expression
1		nfcv 2372	. 2 |- F/_ x A
2		nfv 1574	. 2 |- F/ x ps
3		vtoclga.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4		vtoclga.2	. 2 |- ( x e. B -> ph )
5	1, 2, 3, 4	vtoclgaf 2866	1 |- ( A e. B -> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  vtoclri  2878  ssuni  3910  ordtriexmid  4613  onsucsssucexmid  4619  tfis3  4678  fvmpt3  5713  fvmptssdm  5719  fnressn  5825  fressnfv  5826  caovord  6177  caovimo  6199  tfrlem1  6454  nnacl  6626  nnmcl  6627  nnacom  6630  nnaass  6631  nndi  6632  nnmass  6633  nnmsucr  6634  nnmcom  6635  nnsucsssuc  6638  nntri3or  6639  nnaordi  6654  nnaword  6657  nnmordi  6662  nnaordex  6674  ixpfn  6851  findcard  7050  findcard2  7051  findcard2s  7052  exmidomni  7309  indpi  7529  prarloclem3  7684  uzind4s2  9786  cnref1o  9846  frec2uzrdg  10631  expcl2lemap  10773  seq3coll  11064  climub  11855  climserle  11856  fsum3cvg  11889  summodclem2a  11892  prodfap0  12056  prodfrecap  12057  fproddccvg  12083  alginv  12569  algcvg  12570  algcvga  12573  algfx  12574  prmind2  12642  prmpwdvds  12878  lgsdir2lem4  15710