Theorem pwjust 4159

Description: Soundness justification theorem for df-pw 4160. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
pwjust	|- { x | x C_ A } = { y | y C_ A }

Distinct variable groups:   x, A   y, A

Proof of Theorem pwjust

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3626	. . 3 |- ( x = z -> ( x C_ A <-> z C_ A ) )
2	1	cbvabv 2747	. 2 |- { x | x C_ A } = { z | z C_ A }
3		sseq1 3626	. . 3 |- ( z = y -> ( z C_ A <-> y C_ A ) )
4	3	cbvabv 2747	. 2 |- { z | z C_ A } = { y | y C_ A }
5	2, 4	eqtri 2644	1 |- { x | x C_ A } = { y | y C_ A }

Syntax hints:   = wceq 1483   {cab 2608   C_ wss 3574

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588

This theorem is referenced by: (None)