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Theorem iuneq1d 3894
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 3884 . 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 1348   U_ciun 3871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-iun 3873
This theorem is referenced by:  iuneq12d  3895
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