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Theorem iuneq2f 35315
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 2988 . 2 𝑥 𝐴 = 𝐵
4 id 22 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
5 eqidd 2819 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
63, 1, 2, 4, 5iuneq12df 4936 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wnfc 2958   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-rex 3141  df-iun 4912
This theorem is referenced by:  iuneq12f  35322
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