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Theorem iunconst 4495
Description: Indexed union of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunconst (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iunconst
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.9rzv 4037 . . 3 (𝐴 ≠ ∅ → (𝑦𝐵 ↔ ∃𝑥𝐴 𝑦𝐵))
2 eliun 4490 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
31, 2syl6rbbr 279 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
43eqrdv 2619 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wne 2790  wrex 2908  c0 3891   ciun 4485
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-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-nul 3892  df-iun 4487
This theorem is referenced by:  iununi  4576  oe1m  7570  oarec  7587  oelim2  7620  bnj1143  30566  poimirlem32  33070  mblfinlem2  33076  hoicvr  40066  ovnlecvr2  40128  iunhoiioo  40194
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