Theorem pwjust 3964

Description: Soundness justification theorem for df-pw 3965. (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 3465	. . 3 |- ( x = z -> ( x C_ A <-> z C_ A ) )
2	1	cbvabv 2586	. 2 |- { x | x C_ A } = { z | z C_ A }
3		sseq1 3465	. . 3 |- ( z = y -> ( z C_ A <-> y C_ A ) )
4	3	cbvabv 2586	. 2 |- { z | z C_ A } = { y | y C_ A }
5	2, 4	eqtri 2484	1 |- { x | x C_ A } = { y | y C_ A }

Syntax hints:   = wceq 1455   {cab 2448   C_ wss 3416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-in 3423  df-ss 3430

This theorem is referenced by: (None)