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Theorem iuneq12f 33142
 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 4473 . 2 (∀𝑥𝐴 𝐶 = 𝐷 𝑥𝐴 𝐶 = 𝑥𝐴 𝐷)
2 iuneq12f.1 . . 3 𝑥𝐴
3 iuneq12f.2 . . 3 𝑥𝐵
42, 3iuneq2f 33133 . 2 (𝐴 = 𝐵 𝑥𝐴 𝐷 = 𝑥𝐵 𝐷)
51, 4sylan9eqr 2666 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnfc 2738  ∀wral 2896  ∪ ciun 4455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-iun 4457 This theorem is referenced by: (None)
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