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Theorem iuneq2d 3755
Description: Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
Hypothesis
Ref Expression
iuneq2d.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
iuneq2d (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iuneq2d
StepHypRef Expression
1 iuneq2d.2 . . 3 (𝜑𝐵 = 𝐶)
21adantr 270 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32iuneq2dv 3751 1 (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438   ciun 3730
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 3005  df-ss 3012  df-iun 3732
This theorem is referenced by:  rdgeq1  6136
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