Theorem bj-vjust 32786

Description: Remove dependency on ax-13 2246 from vjust 3201 (note the absence of DV conditions). Soundness justification theorem for df-v 3202. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-vjust

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equid 1939	. . . . 5 |- x = x
2	1	bj-sbtv 32766	. . . 4 |- [ z / x ] x = x
3		equid 1939	. . . . 5 |- y = y
4	3	bj-sbtv 32766	. . . 4 |- [ z / y ] y = y
5	2, 4	2th 254	. . 3 |- ( [ z / x ] x = x <-> [ z / y ] y = y )
6		df-clab 2609	. . 3 |- ( z e. { x | x = x } <-> [ z / x ] x = x )
7		df-clab 2609	. . 3 |- ( z e. { y | y = y } <-> [ z / y ] y = y )
8	5, 6, 7	3bitr4i 292	. 2 |- ( z e. { x | x = x } <-> z e. { y | y = y } )
9	8	eqriv 2619	1 |- { x | x = x } = { y | y = y }

Syntax hints:   = wceq 1483   [wsb 1880   e. wcel 1990   {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615

This theorem is referenced by: (None)