Theorem cbviuneq12df 37953

Description: Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)

Hypotheses

Ref	Expression
cbviuneq12df.xph	|- F/ x ph
cbviuneq12df.yph	|- F/ y ph
cbviuneq12df.x	|- F/_ x X
cbviuneq12df.y	|- F/_ y Y
cbviuneq12df.xa	|- F/_ x A
cbviuneq12df.ya	|- F/_ y A
cbviuneq12df.b	|- F/_ y B
cbviuneq12df.xc	|- F/_ x C
cbviuneq12df.yc	|- F/_ y C
cbviuneq12df.d	|- F/_ x D
cbviuneq12df.f	|- F/_ x F
cbviuneq12df.g	|- F/_ y G
cbviuneq12df.xel	|- ( ( ph /\ y e. C ) -> X e. A )
cbviuneq12df.yel	|- ( ( ph /\ x e. A ) -> Y e. C )
cbviuneq12df.xsub	|- ( ( ph /\ y e. C /\ x = X ) -> B = F )
cbviuneq12df.ysub	|- ( ( ph /\ x e. A /\ y = Y ) -> D = G )
cbviuneq12df.eq1	|- ( ( ph /\ x e. A ) -> B = G )
cbviuneq12df.eq2	|- ( ( ph /\ y e. C ) -> D = F )

Assertion

Ref	Expression
cbviuneq12df	|- ( ph -> U_ x e. A B = U_ y e. C D )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   A( x, y)   B( x, y)   C( x, y)   D( x, y)   F( x, y)   G( x, y)   X( x, y)   Y( x, y)

Proof of Theorem cbviuneq12df

Step	Hyp	Ref	Expression
1		cbviuneq12df.xph	. . 3 |- F/ x ph
2		cbviuneq12df.yph	. . 3 |- F/ y ph
3		cbviuneq12df.y	. . 3 |- F/_ y Y
4		cbviuneq12df.ya	. . 3 |- F/_ y A
5		cbviuneq12df.b	. . 3 |- F/_ y B
6		cbviuneq12df.xc	. . 3 |- F/_ x C
7		cbviuneq12df.yc	. . 3 |- F/_ y C
8		cbviuneq12df.d	. . 3 |- F/_ x D
9		cbviuneq12df.g	. . 3 |- F/_ y G
10		cbviuneq12df.yel	. . 3 |- ( ( ph /\ x e. A ) -> Y e. C )
11		cbviuneq12df.ysub	. . 3 |- ( ( ph /\ x e. A /\ y = Y ) -> D = G )
12		cbviuneq12df.eq1	. . . 4 |- ( ( ph /\ x e. A ) -> B = G )
13		eqimss 3657	. . . 4 |- ( B = G -> B C_ G )
14	12, 13	syl 17	. . 3 |- ( ( ph /\ x e. A ) -> B C_ G )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14	ss2iundf 37951	. 2 |- ( ph -> U_ x e. A B C_ U_ y e. C D )
16		cbviuneq12df.x	. . 3 |- F/_ x X
17		cbviuneq12df.xa	. . 3 |- F/_ x A
18		cbviuneq12df.f	. . 3 |- F/_ x F
19		cbviuneq12df.xel	. . 3 |- ( ( ph /\ y e. C ) -> X e. A )
20		cbviuneq12df.xsub	. . 3 |- ( ( ph /\ y e. C /\ x = X ) -> B = F )
21		cbviuneq12df.eq2	. . . 4 |- ( ( ph /\ y e. C ) -> D = F )
22		eqimss 3657	. . . 4 |- ( D = F -> D C_ F )
23	21, 22	syl 17	. . 3 |- ( ( ph /\ y e. C ) -> D C_ F )
24	2, 1, 16, 6, 8, 4, 17, 5, 18, 19, 20, 23	ss2iundf 37951	. 2 |- ( ph -> U_ y e. C D C_ U_ x e. A B )
25	15, 24	eqssd 3620	1 |- ( ph -> U_ x e. A B = U_ y e. C D )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   C_ wss 3574   U_ciun 4520

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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by:  cbviuneq12dv  37954