Theorem rabxfr 4202

Description: Class builder membership after substituting an expression A (containing y) for x in the class expression ph. (Contributed by NM, 10-Jun-2005.)

Hypotheses

Ref	Expression
rabxfr.1	|- F/_ y B
rabxfr.2	|- F/_ y C
rabxfr.3	|- ( y e. D -> A e. D )
rabxfr.4	|- ( x = A -> ( ph <-> ps ) )
rabxfr.5	|- ( y = B -> A = C )

Assertion

Ref	Expression
rabxfr	|- ( B e. D -> ( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) )

Distinct variable groups:   x, A   x, y, D   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)   A( y)   B( x, y)   C( x, y)

Proof of Theorem rabxfr

Step	Hyp	Ref	Expression
1		tru 1247	. 2 |- T.
2		rabxfr.1	. . 3 |- F/_ y B
3		rabxfr.2	. . 3 |- F/_ y C
4		rabxfr.3	. . . 4 |- ( y e. D -> A e. D )
5	4	adantl 262	. . 3 |- ( ( T. /\ y e. D ) -> A e. D )
6		rabxfr.4	. . 3 |- ( x = A -> ( ph <-> ps ) )
7		rabxfr.5	. . 3 |- ( y = B -> A = C )
8	2, 3, 5, 6, 7	rabxfrd 4201	. 2 |- ( ( T. /\ B e. D ) -> ( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) )
9	1, 8	mpan 400	1 |- ( B e. D -> ( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   T. wtru 1244   e. wcel 1393   F/_wnfc 2165   {crab 2310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559

This theorem is referenced by: (None)