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Theorem iuneq2f 35552
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 2992 . 2 𝑥 𝐴 = 𝐵
4 id 22 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
5 eqidd 2823 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
63, 1, 2, 4, 5iuneq12df 4920 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wnfc 2960   ciun 4894
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
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 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rex 3136  df-iun 4896
This theorem is referenced by:  iuneq12f  35559
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