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Theorem iuneq2d 3679
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 261 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32iuneq2dv 3675 1 (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393   ciun 3654
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 2308  df-rex 2309  df-v 2556  df-in 2921  df-ss 2928  df-iun 3656
This theorem is referenced by:  rdgeq1  5945
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