Theorem pwjust 3560

Description: Soundness justification theorem for df-pw 3561. (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 3165	. . 3 |- ( x = z -> ( x C_ A <-> z C_ A ) )
2	1	cbvabv 2291	. 2 |- { x | x C_ A } = { z | z C_ A }
3		sseq1 3165	. . 3 |- ( z = y -> ( z C_ A <-> y C_ A ) )
4	3	cbvabv 2291	. 2 |- { z | z C_ A } = { y | y C_ A }
5	2, 4	eqtri 2186	1 |- { x | x C_ A } = { y | y C_ A }

Syntax hints:   = wceq 1343   {cab 2151   C_ wss 3116

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by: (None)