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Theorem iuneq2f 34093
Description: Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
Hypotheses
Ref Expression
iuneq2f.1 𝑥𝐴
iuneq2f.2 𝑥𝐵
Assertion
Ref Expression
iuneq2f (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Proof of Theorem iuneq2f
StepHypRef Expression
1 iuneq2f.1 . . 3 𝑥𝐴
2 iuneq2f.2 . . 3 𝑥𝐵
31, 2nfeq 2805 . 2 𝑥 𝐴 = 𝐵
4 id 22 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
5 eqidd 2652 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
63, 1, 2, 4, 5iuneq12df 4576 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wnfc 2780   ciun 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-iun 4554
This theorem is referenced by:  iuneq12f  34102
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