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Theorem iunconst 4720
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 4259 . . 3 (𝐴 ≠ ∅ → (𝑦𝐵 ↔ ∃𝑥𝐴 𝑦𝐵))
2 eliun 4715 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
31, 2syl6rbbr 282 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
43eqrdv 2798 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  wne 2972  wrex 3091  c0 4116   ciun 4711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-v 3388  df-dif 3773  df-nul 4117  df-iun 4713
This theorem is referenced by:  iununi  4802  oe1m  7866  oarec  7883  oelim2  7916  bnj1143  31377  poimirlem32  33929  mblfinlem2  33935  hoicvr  41503  ovnlecvr2  41565  iunhoiioo  41631
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