Theorem vjust 2777

Description: Soundness justification theorem for df-v 2778. (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 1725	. . . . 5 |- x = x
2	1	sbt 1808	. . . 4 |- [ z / x ] x = x
3		equid 1725	. . . . 5 |- y = y
4	3	sbt 1808	. . . 4 |- [ z / y ] y = y
5	2, 4	2th 174	. . 3 |- ( [ z / x ] x = x <-> [ z / y ] y = y )
6		df-clab 2194	. . 3 |- ( z e. { x | x = x } <-> [ z / x ] x = x )
7		df-clab 2194	. . 3 |- ( z e. { y | y = y } <-> [ z / y ] y = y )
8	5, 6, 7	3bitr4i 212	. 2 |- ( z e. { x | x = x } <-> z e. { y | y = y } )
9	8	eqriv 2204	1 |- { x | x = x } = { y | y = y }

Syntax hints:   = wceq 1373   [wsb 1786   e. wcel 2178   {cab 2193

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-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200

This theorem is referenced by: (None)