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Theorem iuneq12f 29125
Description: Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Hypotheses
Ref Expression
iuneq12f.1  |-  F/_ x A
iuneq12f.2  |-  F/_ x B
Assertion
Ref Expression
iuneq12f  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  D )  ->  U_ x  e.  A  C  =  U_ x  e.  B  D
)

Proof of Theorem iuneq12f
StepHypRef Expression
1 iuneq2 4296 . 2  |-  ( A. x  e.  A  C  =  D  ->  U_ x  e.  A  C  =  U_ x  e.  A  D
)
2 iuneq12f.1 . . 3  |-  F/_ x A
3 iuneq12f.2 . . 3  |-  F/_ x B
42, 3iuneq2f 29116 . 2  |-  ( A  =  B  ->  U_ x  e.  A  D  =  U_ x  e.  B  D
)
51, 4sylan9eqr 2517 1  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  D )  ->  U_ x  e.  A  C  =  U_ x  e.  B  D
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   F/_wnfc 2602   A.wral 2799   U_ciun 4280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-in 3444  df-ss 3451  df-iun 4282
This theorem is referenced by: (None)
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