Theorem qseq2 6602

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 6590	. . . . 5 |- ( A = B -> [ x ] A = [ x ] B )
2	1	eqeq2d 2201	. . . 4 |- ( A = B -> ( y = [ x ] A <-> y = [ x ] B ) )
3	2	rexbidv 2491	. . 3 |- ( A = B -> ( E. x e. C y = [ x ] A <-> E. x e. C y = [ x ] B ) )
4	3	abbidv 2307	. 2 |- ( A = B -> { y | E. x e. C y = [ x ] A } = { y | E. x e. C y = [ x ] B } )
5		df-qs 6559	. 2 |- ( C /. A ) = { y | E. x e. C y = [ x ] A }
6		df-qs 6559	. 2 |- ( C /. B ) = { y | E. x e. C y = [ x ] B }
7	4, 5, 6	3eqtr4g 2247	1 |- ( A = B -> ( C /. A ) = ( C /. B ) )

Syntax hints:   -> wi 4   = wceq 1364   {cab 2175   E.wrex 2469   [cec 6551   /.cqs 6552

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-cnv 4649  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-ec 6555  df-qs 6559

This theorem is referenced by: (None)