Theorem qseq2 6529

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 6517	. . . . 5 |- ( A = B -> [ x ] A = [ x ] B )
2	1	eqeq2d 2169	. . . 4 |- ( A = B -> ( y = [ x ] A <-> y = [ x ] B ) )
3	2	rexbidv 2458	. . 3 |- ( A = B -> ( E. x e. C y = [ x ] A <-> E. x e. C y = [ x ] B ) )
4	3	abbidv 2275	. 2 |- ( A = B -> { y | E. x e. C y = [ x ] A } = { y | E. x e. C y = [ x ] B } )
5		df-qs 6486	. 2 |- ( C /. A ) = { y | E. x e. C y = [ x ] A }
6		df-qs 6486	. 2 |- ( C /. B ) = { y | E. x e. C y = [ x ] B }
7	4, 5, 6	3eqtr4g 2215	1 |- ( A = B -> ( C /. A ) = ( C /. B ) )

Syntax hints:   -> wi 4   = wceq 1335   {cab 2143   E.wrex 2436   [cec 6478   /.cqs 6479

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-ec 6482  df-qs 6486

This theorem is referenced by: (None)