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Theorem iuneq12d 3929
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
StepHypRef Expression
1 iuneq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21iuneq1d 3928 . 2  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  C )
3 iuneq12d.2 . . . 4  |-  ( ph  ->  C  =  D )
43adantr 451 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  C  =  D )
54iuneq2dv 3926 . 2  |-  ( ph  ->  U_ x  e.  B  C  =  U_ x  e.  B  D )
62, 5eqtrd 2315 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   U_ciun 3905
This theorem is referenced by:  cfsmolem  7896  cfsmo  7897  wunex2  8360  wuncval2  8369  imasval  13414  lpival  15997  dvfval  19247  heiborlem10  26544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-iun 3907
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