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Theorem iuneq12d 3749
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 3748 . 2 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
3 iuneq12d.2 . . . 4 (𝜑𝐶 = 𝐷)
43adantr 270 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
54iuneq2dv 3746 . 2 (𝜑 𝑥𝐵 𝐶 = 𝑥𝐵 𝐷)
62, 5eqtrd 2120 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438   ciun 3725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3003  df-ss 3010  df-iun 3727
This theorem is referenced by:  rdgivallem  6128  rdgon  6133  rdg0  6134
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