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Theorem iuneq12f 35322
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 4929 . 2 (∀𝑥𝐴 𝐶 = 𝐷 𝑥𝐴 𝐶 = 𝑥𝐴 𝐷)
2 iuneq12f.1 . . 3 𝑥𝐴
3 iuneq12f.2 . . 3 𝑥𝐵
42, 3iuneq2f 35315 . 2 (𝐴 = 𝐵 𝑥𝐴 𝐷 = 𝑥𝐵 𝐷)
51, 4sylan9eqr 2875 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wnfc 2958  wral 3135   ciun 4910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-in 3940  df-ss 3949  df-iun 4912
This theorem is referenced by: (None)
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