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Theorem iuneq12f 35668
 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 4903 . 2 (∀𝑥𝐴 𝐶 = 𝐷 𝑥𝐴 𝐶 = 𝑥𝐴 𝐷)
2 iuneq12f.1 . . 3 𝑥𝐴
3 iuneq12f.2 . . 3 𝑥𝐵
42, 3iuneq2f 35661 . 2 (𝐴 = 𝐵 𝑥𝐴 𝐷 = 𝑥𝐵 𝐷)
51, 4sylan9eqr 2855 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  Ⅎwnfc 2936  ∀wral 3106  ∪ ciun 4884 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3889  df-ss 3899  df-iun 4886 This theorem is referenced by: (None)
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