Theorem iuneq12d 4546

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 4545	. 2 |- ( ph -> U_ x e. A C = U_ x e. B C )
3		iuneq12d.2	. . . 4 |- ( ph -> C = D )
4	3	adantr 481	. . 3 |- ( ( ph /\ x e. B ) -> C = D )
5	4	iuneq2dv 4542	. 2 |- ( ph -> U_ x e. B C = U_ x e. B D )
6	2, 5	eqtrd 2656	1 |- ( ph -> U_ x e. A C = U_ x e. B D )

Syntax hints:   -> wi 4   = 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-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:  disjiunb  4642  otiunsndisj  4980  cfsmolem  9092  cfsmo  9093  wunex2  9560  wuncval2  9569  s3iunsndisj  13707  imasval  16171  lpival  19245  cnextval  21865  cnextfval  21866  dvfval  23661  mblfinlem2  33447  heiborlem10  33619  iunrelexpmin1  38000  iunrelexpmin2  38004