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Theorem iuneq12f 36611
Description: Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Hypotheses
Ref Expression
iuneq12f.1 𝑥𝐴
iuneq12f.2 𝑥𝐵
Assertion
Ref Expression
iuneq12f ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Proof of Theorem iuneq12f
StepHypRef Expression
1 iuneq2 4972 . 2 (∀𝑥𝐴 𝐶 = 𝐷 𝑥𝐴 𝐶 = 𝑥𝐴 𝐷)
2 iuneq12f.1 . . 3 𝑥𝐴
3 iuneq12f.2 . . 3 𝑥𝐵
42, 3iuneq2f 36604 . 2 (𝐴 = 𝐵 𝑥𝐴 𝐷 = 𝑥𝐵 𝐷)
51, 4sylan9eqr 2798 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wnfc 2886  wral 3063   ciun 4953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ral 3064  df-rex 3073  df-v 3446  df-in 3916  df-ss 3926  df-iun 4955
This theorem is referenced by: (None)
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