Theorem cbviuneq12dv 37954

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
cbviuneq12dv.xel	|- ( ( ph /\ y e. C ) -> X e. A )
cbviuneq12dv.yel	|- ( ( ph /\ x e. A ) -> Y e. C )
cbviuneq12dv.xsub	|- ( ( ph /\ y e. C /\ x = X ) -> B = F )
cbviuneq12dv.ysub	|- ( ( ph /\ x e. A /\ y = Y ) -> D = G )
cbviuneq12dv.eq1	|- ( ( ph /\ x e. A ) -> B = G )
cbviuneq12dv.eq2	|- ( ( ph /\ y e. C ) -> D = F )

Assertion

Ref	Expression
cbviuneq12dv	|- ( ph -> U_ x e. A B = U_ y e. C D )

Distinct variable groups:   x, y, ph   x, A, y   y, B   x, C, y   x, D   x, F   y, G   x, X   y, Y

Allowed substitution hints:   B( x)   D( y)   F( y)   G( x)   X( y)   Y( x)

Proof of Theorem cbviuneq12dv

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ x ph
2		nfv 1843	. 2 |- F/ y ph
3		nfcv 2764	. 2 |- F/_ x X
4		nfcv 2764	. 2 |- F/_ y Y
5		nfcv 2764	. 2 |- F/_ x A
6		nfcv 2764	. 2 |- F/_ y A
7		nfcv 2764	. 2 |- F/_ y B
8		nfcv 2764	. 2 |- F/_ x C
9		nfcv 2764	. 2 |- F/_ y C
10		nfcv 2764	. 2 |- F/_ x D
11		nfcv 2764	. 2 |- F/_ x F
12		nfcv 2764	. 2 |- F/_ y G
13		cbviuneq12dv.xel	. 2 |- ( ( ph /\ y e. C ) -> X e. A )
14		cbviuneq12dv.yel	. 2 |- ( ( ph /\ x e. A ) -> Y e. C )
15		cbviuneq12dv.xsub	. 2 |- ( ( ph /\ y e. C /\ x = X ) -> B = F )
16		cbviuneq12dv.ysub	. 2 |- ( ( ph /\ x e. A /\ y = Y ) -> D = G )
17		cbviuneq12dv.eq1	. 2 |- ( ( ph /\ x e. A ) -> B = G )
18		cbviuneq12dv.eq2	. 2 |- ( ( ph /\ y e. C ) -> D = F )
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	cbviuneq12df 37953	1 |- ( ph -> U_ x e. A B = U_ y e. C D )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   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:  trclfvdecomr  38020