Theorem iuneq12d 3874

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 3873	. 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 3871	. 2 |- ( ph -> U_ x e. B C = U_ x e. B D )
6	2, 5	eqtrd 2190	1 |- ( ph -> U_ x e. A C = U_ x e. B D )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   U_ciun 3850

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-iun 3852

This theorem is referenced by:  rdgivallem  6329  rdgon  6334  rdg0  6335  reldvg  13090  dvfvalap  13092