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Theorem iuneq12d 4514
 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 4513 . 2 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
3 iuneq12d.2 . . . 4 (𝜑𝐶 = 𝐷)
43adantr 481 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
54iuneq2dv 4510 . 2 (𝜑 𝑥𝐵 𝐶 = 𝑥𝐵 𝐷)
62, 5eqtrd 2655 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ∪ ciun 4487 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-in 3563  df-ss 3570  df-iun 4489 This theorem is referenced by:  otiunsndisj  4942  cfsmolem  9039  cfsmo  9040  wunex2  9507  wuncval2  9516  s3iunsndisj  13644  imasval  16095  lpival  19167  cnextval  21778  cnextfval  21779  dvfval  23574  2wspiundisj  26731  mblfinlem2  33100  heiborlem10  33272  iunrelexpmin1  37502  iunrelexpmin2  37506
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