Theorem iuneq12d 3965

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 3964	. 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 3962	. 2 |- ( ph -> U_ x e. B C = U_ x e. B D )
6	2, 5	eqtrd 2240	1 |- ( ph -> U_ x e. A C = U_ x e. B D )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   U_ciun 3941

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-iun 3943

This theorem is referenced by:  rdgivallem  6490  rdgon  6495  rdg0  6496  imasival  13253  reldvg  15266  dvfvalap  15268