Theorem unjust 3119

Description: Soundness justification theorem for df-un 3120. (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 2229	. . . 4 |- ( x = z -> ( x e. A <-> z e. A ) )
2		eleq1 2229	. . . 4 |- ( x = z -> ( x e. B <-> z e. B ) )
3	1, 2	orbi12d 783	. . 3 |- ( x = z -> ( ( x e. A \/ x e. B ) <-> ( z e. A \/ z e. B ) ) )
4	3	cbvabv 2291	. 2 |- { x | ( x e. A \/ x e. B ) } = { z | ( z e. A \/ z e. B ) }
5		eleq1 2229	. . . 4 |- ( z = y -> ( z e. A <-> y e. A ) )
6		eleq1 2229	. . . 4 |- ( z = y -> ( z e. B <-> y e. B ) )
7	5, 6	orbi12d 783	. . 3 |- ( z = y -> ( ( z e. A \/ z e. B ) <-> ( y e. A \/ y e. B ) ) )
8	7	cbvabv 2291	. 2 |- { z | ( z e. A \/ z e. B ) } = { y | ( y e. A \/ y e. B ) }
9	4, 8	eqtri 2186	1 |- { x | ( x e. A \/ x e. B ) } = { y | ( y e. A \/ y e. B ) }

Syntax hints:   \/ wo 698   = wceq 1343   e. wcel 2136   {cab 2151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161

This theorem is referenced by: (None)