Theorem pwjust 3602

Description: Soundness justification theorem for df-pw 3603. (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 3202	. . 3 |- ( x = z -> ( x C_ A <-> z C_ A ) )
2	1	cbvabv 2318	. 2 |- { x | x C_ A } = { z | z C_ A }
3		sseq1 3202	. . 3 |- ( z = y -> ( z C_ A <-> y C_ A ) )
4	3	cbvabv 2318	. 2 |- { z | z C_ A } = { y | y C_ A }
5	2, 4	eqtri 2214	1 |- { x | x C_ A } = { y | y C_ A }

Syntax hints:   = wceq 1364   {cab 2179   C_ wss 3153

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by: (None)