Theorem iuneq12d 3936

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 3935	. 2 |- ( ph -> U_ x e. A C = U_ x e. B C )
3		iuneq12d.2	. . . 4 |- ( ph -> C = D )
4	3	adantr 276	. . 3 |- ( ( ph /\ x e. B ) -> C = D )
5	4	iuneq2dv 3933	. 2 |- ( ph -> U_ x e. B C = U_ x e. B D )
6	2, 5	eqtrd 2226	1 |- ( ph -> U_ x e. A C = U_ x e. B D )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   U_ciun 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-iun 3914

This theorem is referenced by:  rdgivallem  6425  rdgon  6430  rdg0  6431  imasival  12879  reldvg  14810  dvfvalap  14812