Theorem pwjust 3627

Description: Soundness justification theorem for df-pw 3628. (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 3224	. . 3 |- ( x = z -> ( x C_ A <-> z C_ A ) )
2	1	cbvabv 2332	. 2 |- { x | x C_ A } = { z | z C_ A }
3		sseq1 3224	. . 3 |- ( z = y -> ( z C_ A <-> y C_ A ) )
4	3	cbvabv 2332	. 2 |- { z | z C_ A } = { y | y C_ A }
5	2, 4	eqtri 2228	1 |- { x | x C_ A } = { y | y C_ A }

Syntax hints:   = wceq 1373   {cab 2193   C_ wss 3174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187

This theorem is referenced by: (None)