Theorem vtoclbg 2839

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 2838	1 |- ( A e. V -> ( ch <-> th ) )

Syntax hints:   -> wi 4   <-> wb 105   = 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:  pm13.183  2918  sbc8g  3013  sbcco  3027  sbc5  3029  sbcie2g  3039  eqsbc1  3045  sbcng  3046  sbcimg  3047  sbcan  3048  sbcang  3049  sbcor  3050  sbcorg  3051  sbcbig  3052  sbcal  3057  sbcalg  3058  sbcex2  3059  sbcexg  3060  sbcel1v  3068  sbcralg  3084  sbcreug  3086  sbcel12g  3116  sbceqg  3117  csbiebg  3144  elpwg  3634  snssgOLD  3780  preq12bg  3827  elintg  3907  elintrabg  3912  sbcbrg  4114  opelresg  4985  elixpsn  6845  ixpsnf1o  6846  domeng  6864