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Theorem iuneq12f 30778
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 4277 . 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 30769 . 2  |-  ( A  =  B  ->  U_ x  e.  A  D  =  U_ x  e.  B  D
)
51, 4sylan9eqr 2459 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 367    = wceq 1399   F/_wnfc 2544   A.wral 2746   U_ciun 4260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ral 2751  df-rex 2752  df-v 3053  df-in 3413  df-ss 3420  df-iun 4262
This theorem is referenced by: (None)
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