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Theorem iuneq12d 4680
Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
Hypotheses
Ref Expression
iuneq1d.1 (𝜑𝐴 = 𝐵)
iuneq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
iuneq12d (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem iuneq12d
StepHypRef Expression
1 iuneq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21iuneq1d 4679 . 2 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
3 iuneq12d.2 . . . 4 (𝜑𝐶 = 𝐷)
43adantr 466 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
54iuneq2dv 4676 . 2 (𝜑 𝑥𝐵 𝐶 = 𝑥𝐵 𝐷)
62, 5eqtrd 2805 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145   ciun 4654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-iun 4656
This theorem is referenced by:  disjiunb  4776  otiunsndisj  5113  cfsmolem  9294  cfsmo  9295  wunex2  9762  wuncval2  9771  s3iunsndisj  13917  imasval  16379  lpival  19460  cnextval  22085  cnextfval  22086  dvfval  23881  mblfinlem2  33780  heiborlem10  33951  iunrelexpmin1  38526  iunrelexpmin2  38530
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