Theorem qseq2 6471

Description: Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Assertion

Ref	Expression
qseq2	|- ( A = B -> ( C /. A ) = ( C /. B ) )

Proof of Theorem qseq2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eceq2 6459	. . . . 5 |- ( A = B -> [ x ] A = [ x ] B )
2	1	eqeq2d 2149	. . . 4 |- ( A = B -> ( y = [ x ] A <-> y = [ x ] B ) )
3	2	rexbidv 2436	. . 3 |- ( A = B -> ( E. x e. C y = [ x ] A <-> E. x e. C y = [ x ] B ) )
4	3	abbidv 2255	. 2 |- ( A = B -> { y | E. x e. C y = [ x ] A } = { y | E. x e. C y = [ x ] B } )
5		df-qs 6428	. 2 |- ( C /. A ) = { y | E. x e. C y = [ x ] A }
6		df-qs 6428	. 2 |- ( C /. B ) = { y | E. x e. C y = [ x ] B }
7	4, 5, 6	3eqtr4g 2195	1 |- ( A = B -> ( C /. A ) = ( C /. B ) )

Syntax hints:   -> wi 4   = wceq 1331   {cab 2123   E.wrex 2415   [cec 6420   /.cqs 6421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-ec 6424  df-qs 6428

This theorem is referenced by: (None)