Theorem qseq2 7797

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 7784	. . . . 5 |- ( A = B -> [ x ] A = [ x ] B )
2	1	eqeq2d 2632	. . . 4 |- ( A = B -> ( y = [ x ] A <-> y = [ x ] B ) )
3	2	rexbidv 3052	. . 3 |- ( A = B -> ( E. x e. C y = [ x ] A <-> E. x e. C y = [ x ] B ) )
4	3	abbidv 2741	. 2 |- ( A = B -> { y | E. x e. C y = [ x ] A } = { y | E. x e. C y = [ x ] B } )
5		df-qs 7748	. 2 |- ( C /. A ) = { y | E. x e. C y = [ x ] A }
6		df-qs 7748	. 2 |- ( C /. B ) = { y | E. x e. C y = [ x ] B }
7	4, 5, 6	3eqtr4g 2681	1 |- ( A = B -> ( C /. A ) = ( C /. B ) )

Syntax hints:   -> wi 4   = wceq 1483   {cab 2608   E.wrex 2913   [cec 7740   /.cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  pi1bas3  22843  pstmval  29938  qseq2i  34056  qseq2d  34057  qseq12  34058