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Theorem iuneq2f 30156
Description: Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
Hypotheses
Ref Expression
iuneq2f.1  |-  F/_ x A
iuneq2f.2  |-  F/_ x B
Assertion
Ref Expression
iuneq2f  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)

Proof of Theorem iuneq2f
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iuneq2f.1 . . 3  |-  F/_ x A
2 iuneq2f.2 . . 3  |-  F/_ x B
31, 2nfeq 2635 . 2  |-  F/ x  A  =  B
4 eleq2 2535 . . 3  |-  ( A  =  B  ->  (
y  e.  A  <->  y  e.  B ) )
54eqrdv 2459 . 2  |-  ( A  =  B  ->  A  =  B )
6 eqidd 2463 . 2  |-  ( A  =  B  ->  C  =  C )
73, 1, 2, 5, 6iuneq12df 4344 1  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374   F/_wnfc 2610   U_ciun 4320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rex 2815  df-iun 4322
This theorem is referenced by:  iuneq12f  30165
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