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Theorem iuneq12d 3737
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 3736 . 2  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  C )
3 iuneq12d.2 . . . 4  |-  ( ph  ->  C  =  D )
43adantr 270 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  C  =  D )
54iuneq2dv 3734 . 2  |-  ( ph  ->  U_ x  e.  B  C  =  U_ x  e.  B  D )
62, 5eqtrd 2117 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436   U_ciun 3713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-in 2994  df-ss 3001  df-iun 3715
This theorem is referenced by:  rdgivallem  6100  rdgon  6105  rdg0  6106
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