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Theorem iuneq12f 33952
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 4535 . 2 (∀𝑥𝐴 𝐶 = 𝐷 𝑥𝐴 𝐶 = 𝑥𝐴 𝐷)
2 iuneq12f.1 . . 3 𝑥𝐴
3 iuneq12f.2 . . 3 𝑥𝐵
42, 3iuneq2f 33943 . 2 (𝐴 = 𝐵 𝑥𝐴 𝐷 = 𝑥𝐵 𝐷)
51, 4sylan9eqr 2677 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wnfc 2750  wral 2911   ciun 4518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-v 3200  df-in 3579  df-ss 3586  df-iun 4520
This theorem is referenced by: (None)
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