Theorem qseq2 6155

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 6143	. . . . 5 |- ( A = B -> [ x ] A = [ x ] B )
2	1	eqeq2d 2051	. . . 4 |- ( A = B -> ( y = [ x ] A <-> y = [ x ] B ) )
3	2	rexbidv 2327	. . 3 |- ( A = B -> ( E. x e. C y = [ x ] A <-> E. x e. C y = [ x ] B ) )
4	3	abbidv 2155	. 2 |- ( A = B -> { y | E. x e. C y = [ x ] A } = { y | E. x e. C y = [ x ] B } )
5		df-qs 6112	. 2 |- ( C /. A ) = { y | E. x e. C y = [ x ] A }
6		df-qs 6112	. 2 |- ( C /. B ) = { y | E. x e. C y = [ x ] B }
7	4, 5, 6	3eqtr4g 2097	1 |- ( A = B -> ( C /. A ) = ( C /. B ) )

Syntax hints:   -> wi 4   = wceq 1243   {cab 2026   E.wrex 2307   [cec 6104   /.cqs 6105

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-ec 6108  df-qs 6112

This theorem is referenced by: (None)