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Theorem iuneq12d 3681
 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 3680 . 2 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
3 iuneq12d.2 . . . 4 (𝜑𝐶 = 𝐷)
43adantr 261 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
54iuneq2dv 3678 . 2 (𝜑 𝑥𝐵 𝐶 = 𝑥𝐵 𝐷)
62, 5eqtrd 2072 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ∪ ciun 3657 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iun 3659 This theorem is referenced by:  rdgivallem  5968  rdg0  5974
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