Theorem vtoclbg 2875

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 2874	1 |- ( A e. V -> ( ch <-> th ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814

This theorem is referenced by:  pm13.183  2954  sbc8g  3049  sbcco  3063  sbc5  3065  sbcie2g  3075  eqsbc1  3081  sbcng  3082  sbcimg  3083  sbcan  3084  sbcang  3085  sbcor  3086  sbcorg  3087  sbcbig  3088  sbcal  3093  sbcalg  3094  sbcex2  3095  sbcexg  3096  sbcel1v  3104  sbcralg  3120  sbcreug  3122  sbcel12g  3152  sbceqg  3153  csbiebg  3180  elpwg  3676  snssgOLD  3829  preq12bg  3876  elintg  3956  elintrabg  3961  sbcbrg  4163  opelresg  5044  fsn2g  5851  elixpsn  6969  ixpsnf1o  6970  domeng  6988