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Theorem iuneq1d 3748
Description: Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
Hypothesis
Ref Expression
iuneq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
iuneq1d  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    C( x)

Proof of Theorem iuneq1d
StepHypRef Expression
1 iuneq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 iuneq1 3738 . 2  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)
31, 2syl 14 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   U_ciun 3725
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3003  df-ss 3010  df-iun 3727
This theorem is referenced by:  iuneq12d  3749
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