Theorem rabxfr 4573

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 1402	. 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 277	. . 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 4572	. 2 |- ( ( T. /\ B e. D ) -> ( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) )
9	1, 8	mpan 424	1 |- ( B e. D -> ( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   T. wtru 1399   e. wcel 2202   F/_wnfc 2362   {crab 2515

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-v 2805

This theorem is referenced by: (None)