Theorem vtoclbg 2878

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)

Hypotheses

Ref	Expression
vtoclbg.1	|- ( x = A -> ( ph <-> ch ) )
vtoclbg.2	|- ( x = A -> ( ps <-> th ) )
vtoclbg.3	|- ( ph <-> ps )

Assertion

Ref	Expression
vtoclbg	|- ( A e. V -> ( ch <-> th ) )

Distinct variable groups:   x, A   ch, x   th, x

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

Proof of Theorem vtoclbg

Step	Hyp	Ref	Expression
1		vtoclbg.1	. . 3 |- ( x = A -> ( ph <-> ch ) )
2		vtoclbg.2	. . 3 |- ( x = A -> ( ps <-> th ) )
3	1, 2	bibi12d 235	. 2 |- ( x = A -> ( ( ph <-> ps ) <-> ( ch <-> th ) ) )
4		vtoclbg.3	. 2 |- ( ph <-> ps )
5	3, 4	vtoclg 2877	1 |- ( A e. V -> ( ch <-> th ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2205

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817

This theorem is referenced by:  pm13.183  2958  sbc8g  3053  sbcco  3067  sbc5  3069  sbcie2g  3079  eqsbc1  3085  sbcng  3086  sbcimg  3087  sbcan  3088  sbcang  3089  sbcor  3090  sbcorg  3091  sbcbig  3092  sbcal  3097  sbcalg  3098  sbcex2  3099  sbcexg  3100  sbcel1v  3108  sbcralg  3124  sbcreug  3126  sbcel12g  3156  sbceqg  3157  csbiebg  3184  elpwg  3683  snssgOLD  3836  preq12bg  3883  elintg  3963  elintrabg  3968  sbcbrg  4170  opelresg  5051  fsn2g  5858  elixpsn  6984  ixpsnf1o  6985  domeng  7003