Theorem vjust 2539

Description: Soundness justification theorem for df-v 2540. (Contributed by Rodolfo Medina, 27-Apr-2010.)

Assertion

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

Proof of Theorem vjust

Step	Hyp	Ref	Expression
1		equid 1766	. . . . 5 |- x = x
2	1	sbt 1836	. . . 4 |- [z / x]x = x
3		equid 1766	. . . . 5 |- y = y
4	3	sbt 1836	. . . 4 |- [z / y]y = y
5	2, 4	2th 1026	. . 3 |- ([z / x]x = x <-> [z / y]y = y)
6		df-clab 2129	. . 3 |- (z e. {x | x = x} <-> [z / x]x = x)
7		df-clab 2129	. . 3 |- (z e. {y | y = y} <-> [z / y]y = y)
8	5, 6, 7	3bitr4i 295	. 2 |- (z e. {x | x = x} <-> z e. {y | y = y})
9	8	eqriv 2138	1 |- {x | x = x} = {y | y = y}

Syntax hints:   = wceq 1586   e. wcel 1588  [wsbc 1814  {cab 2128

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1593  ax-12 1598  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-ext 2123

This theorem depends on definitions:  df-bi 220  df-an 339  df-ex 1616  df-sb 1816  df-clab 2129  df-cleq 2134

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