Theorem unjust 3578

Description: Soundness justification theorem for df-un 3579. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
unjust	|- { x | ( x e. A \/ x e. B ) } = { y | ( y e. A \/ y e. B ) }

Distinct variable groups:   x, A   x, B   y, A   y, B

Proof of Theorem unjust

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . 4 |- ( x = z -> ( x e. A <-> z e. A ) )
2		eleq1 2689	. . . 4 |- ( x = z -> ( x e. B <-> z e. B ) )
3	1, 2	orbi12d 746	. . 3 |- ( x = z -> ( ( x e. A \/ x e. B ) <-> ( z e. A \/ z e. B ) ) )
4	3	cbvabv 2747	. 2 |- { x | ( x e. A \/ x e. B ) } = { z | ( z e. A \/ z e. B ) }
5		eleq1 2689	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
6		eleq1 2689	. . . 4 |- ( z = y -> ( z e. B <-> y e. B ) )
7	5, 6	orbi12d 746	. . 3 |- ( z = y -> ( ( z e. A \/ z e. B ) <-> ( y e. A \/ y e. B ) ) )
8	7	cbvabv 2747	. 2 |- { z | ( z e. A \/ z e. B ) } = { y | ( y e. A \/ y e. B ) }
9	4, 8	eqtri 2644	1 |- { x | ( x e. A \/ x e. B ) } = { y | ( y e. A \/ y e. B ) }

Syntax hints:   \/ wo 383   = wceq 1483   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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618

This theorem is referenced by: (None)