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Theorem iuneq2f 32999
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 2666 . 2 𝑥 𝐴 = 𝐵
4 id 22 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
5 eqidd 2515 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
63, 1, 2, 4, 5iuneq12df 4378 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wnfc 2642   ciun 4353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rex 2806  df-iun 4355
This theorem is referenced by:  iuneq12f  33008
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