Theorem rabxfr 4890

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 1487	. 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 482	. . 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 4889	. 2 |- ( ( T. /\ B e. D ) -> ( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) )
9	1, 8	mpan 706	1 |- ( B e. D -> ( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   T. wtru 1484   e. wcel 1990   F/_wnfc 2751   {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202

This theorem is referenced by: (None)