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Theorem iuneq2f 31826
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
StepHypRef Expression
1 iuneq2f.1 . . 3  |-  F/_ x A
2 iuneq2f.2 . . 3  |-  F/_ x B
31, 2nfeq 2575 . 2  |-  F/ x  A  =  B
4 id 22 . 2  |-  ( A  =  B  ->  A  =  B )
5 eqidd 2403 . 2  |-  ( A  =  B  ->  C  =  C )
63, 1, 2, 4, 5iuneq12df 4294 1  |-  ( A  =  B  ->  U_ x  e.  A  C  =  U_ x  e.  B  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405   F/_wnfc 2550   U_ciun 4270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2759  df-iun 4272
This theorem is referenced by:  iuneq12f  31835
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