Theorem vjust 2690

Description: Soundness justification theorem for df-v 2691. (Contributed by Rodolfo Medina, 27-Apr-2010.)

Assertion

Ref	Expression
vjust	|- { x | x = x } = { y | y = y }

Proof of Theorem vjust

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equid 1678	. . . . 5 |- x = x
2	1	sbt 1758	. . . 4 |- [ z / x ] x = x
3		equid 1678	. . . . 5 |- y = y
4	3	sbt 1758	. . . 4 |- [ z / y ] y = y
5	2, 4	2th 173	. . 3 |- ( [ z / x ] x = x <-> [ z / y ] y = y )
6		df-clab 2127	. . 3 |- ( z e. { x | x = x } <-> [ z / x ] x = x )
7		df-clab 2127	. . 3 |- ( z e. { y | y = y } <-> [ z / y ] y = y )
8	5, 6, 7	3bitr4i 211	. 2 |- ( z e. { x | x = x } <-> z e. { y | y = y } )
9	8	eqriv 2137	1 |- { x | x = x } = { y | y = y }

Syntax hints:   = wceq 1332   e. wcel 1481   [wsb 1736   {cab 2126

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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133

This theorem is referenced by: (None)