Theorem vtoclbg 2750

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 234	. 2 |- ( x = A -> ( ( ph <-> ps ) <-> ( ch <-> th ) ) )
4		vtoclbg.3	. 2 |- ( ph <-> ps )
5	3, 4	vtoclg 2749	1 |- ( A e. V -> ( ch <-> th ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691

This theorem is referenced by:  pm13.183  2826  sbc8g  2920  sbcco  2934  sbc5  2936  sbcie2g  2946  eqsbc3  2952  sbcng  2953  sbcimg  2954  sbcan  2955  sbcang  2956  sbcor  2957  sbcorg  2958  sbcbig  2959  sbcal  2964  sbcalg  2965  sbcex2  2966  sbcexg  2967  sbcel1v  2975  sbcralg  2991  sbcreug  2993  sbcel12g  3022  sbceqg  3023  csbiebg  3047  elpwg  3523  snssg  3664  preq12bg  3708  elintg  3787  elintrabg  3792  sbcbrg  3990  opelresg  4834  elixpsn  6637  ixpsnf1o  6638  domeng  6654