Theorem vtoclbg 2822

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 2821	1 |- ( A e. V -> ( ch <-> th ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762

This theorem is referenced by:  pm13.183  2899  sbc8g  2994  sbcco  3008  sbc5  3010  sbcie2g  3020  eqsbc1  3026  sbcng  3027  sbcimg  3028  sbcan  3029  sbcang  3030  sbcor  3031  sbcorg  3032  sbcbig  3033  sbcal  3038  sbcalg  3039  sbcex2  3040  sbcexg  3041  sbcel1v  3049  sbcralg  3065  sbcreug  3067  sbcel12g  3096  sbceqg  3097  csbiebg  3124  elpwg  3610  snssgOLD  3755  preq12bg  3800  elintg  3879  elintrabg  3884  sbcbrg  4084  opelresg  4950  elixpsn  6791  ixpsnf1o  6792  domeng  6808