Theorem pwjust 3724

Description: Soundness justification theorem for df-pw 3725. (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 3293	. . 3 |- (x = z -> (x C_ A <-> z C_ A))
2	1	cbvabv 2473	. 2 |- {x | x C_ A} = {z | z C_ A}
3		sseq1 3293	. . 3 |- (z = y -> (z C_ A <-> y C_ A))
4	3	cbvabv 2473	. 2 |- {z | z C_ A} = {y | y C_ A}
5	2, 4	eqtri 2373	1 |- {x | x C_ A} = {y | y C_ A}

Syntax hints:   = wceq 1642  {cab 2339   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)