Theorem rabxfr 3843

Description: Class builder membership after substituting an expression A (containing y) for x in the class expression ph.

Hypotheses

Ref	Expression
rabxfr.1	|- (z e. B -> A.y z e. B)
rabxfr.2	|- (z e. C -> A.y z e. 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   z,B   z,C   x,y,z,D   ph,y,z   ps,x,z

Proof of Theorem rabxfr

Step	Hyp	Ref	Expression
1		equid 1484	. 2 |- x = x
2		rabxfr.1	. . 3 |- (z e. B -> A.y z e. B)
3		rabxfr.2	. . 3 |- (z e. C -> A.y z e. C)
4		rabxfr.3	. . . 4 |- (y e. D -> A e. D)
5	4	adantl 424	. . 3 |- ((x = x /\ 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 3842	. 2 |- ((x = x /\ B e. D) -> (C e. {x e. D | ph} <-> B e. {y e. D | ps}))
9	1, 8	mpan 759	1 |- (B e. D -> (C e. {x e. D | ph} <-> B e. {y e. D | ps}))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  {crab 2108

This theorem is referenced by:  reuunixfr 3850  dfuzi 7414

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rab 2112  df-v 2294

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