Theorem vjust 2558

Description: Soundness justification theorem for df-v 2559. (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 1589	. . . . 5 |- x = x
2	1	sbt 1667	. . . 4 |- [ z / x ] x = x
3		equid 1589	. . . . 5 |- y = y
4	3	sbt 1667	. . . 4 |- [ z / y ] y = y
5	2, 4	2th 163	. . 3 |- ( [ z / x ] x = x <-> [ z / y ] y = y )
6		df-clab 2027	. . 3 |- ( z e. { x | x = x } <-> [ z / x ] x = x )
7		df-clab 2027	. . 3 |- ( z e. { y | y = y } <-> [ z / y ] y = y )
8	5, 6, 7	3bitr4i 201	. 2 |- ( z e. { x | x = x } <-> z e. { y | y = y } )
9	8	eqriv 2037	1 |- { x | x = x } = { y | y = y }

Syntax hints:   = wceq 1243   e. wcel 1393   [wsb 1645   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033

This theorem is referenced by: (None)