Theorem iuneq12d 3837

Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Hypotheses

Ref	Expression
iuneq1d.1	|- ( ph -> A = B )
iuneq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
iuneq12d	|- ( ph -> U_ x e. A C = U_ x e. B D )

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

Allowed substitution hints:   C( x)   D( x)

Proof of Theorem iuneq12d

Step	Hyp	Ref	Expression
1		iuneq1d.1	. . 3 |- ( ph -> A = B )
2	1	iuneq1d 3836	. 2 |- ( ph -> U_ x e. A C = U_ x e. B C )
3		iuneq12d.2	. . . 4 |- ( ph -> C = D )
4	3	adantr 274	. . 3 |- ( ( ph /\ x e. B ) -> C = D )
5	4	iuneq2dv 3834	. 2 |- ( ph -> U_ x e. B C = U_ x e. B D )
6	2, 5	eqtrd 2172	1 |- ( ph -> U_ x e. A C = U_ x e. B D )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   U_ciun 3813

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-iun 3815

This theorem is referenced by:  rdgivallem  6278  rdgon  6283  rdg0  6284  reldvg  12817  dvfvalap  12819