Theorem vtoclbg 2800

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 2799	1 |- ( A e. V -> ( ch <-> th ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741

This theorem is referenced by:  pm13.183  2877  sbc8g  2972  sbcco  2986  sbc5  2988  sbcie2g  2998  eqsbc1  3004  sbcng  3005  sbcimg  3006  sbcan  3007  sbcang  3008  sbcor  3009  sbcorg  3010  sbcbig  3011  sbcal  3016  sbcalg  3017  sbcex2  3018  sbcexg  3019  sbcel1v  3027  sbcralg  3043  sbcreug  3045  sbcel12g  3074  sbceqg  3075  csbiebg  3101  elpwg  3585  snssgOLD  3730  preq12bg  3775  elintg  3854  elintrabg  3859  sbcbrg  4059  opelresg  4916  elixpsn  6737  ixpsnf1o  6738  domeng  6754