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Theorem elunif 39489
Description: A version of eluni 4471 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
elunif.1 𝑥𝐴
elunif.2 𝑥𝐵
Assertion
Ref Expression
elunif (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
Distinct variable group:   𝐴,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem elunif
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eluni 4471 . 2 (𝐴 𝐵 ↔ ∃𝑦(𝐴𝑦𝑦𝐵))
2 elunif.1 . . . . 5 𝑥𝐴
3 nfcv 2793 . . . . 5 𝑥𝑦
42, 3nfel 2806 . . . 4 𝑥 𝐴𝑦
5 elunif.2 . . . . 5 𝑥𝐵
63, 5nfel 2806 . . . 4 𝑥 𝑦𝐵
74, 6nfan 1868 . . 3 𝑥(𝐴𝑦𝑦𝐵)
8 nfv 1883 . . 3 𝑦(𝐴𝑥𝑥𝐵)
9 eleq2 2719 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
10 eleq1 2718 . . . 4 (𝑦 = 𝑥 → (𝑦𝐵𝑥𝐵))
119, 10anbi12d 747 . . 3 (𝑦 = 𝑥 → ((𝐴𝑦𝑦𝐵) ↔ (𝐴𝑥𝑥𝐵)))
127, 8, 11cbvex 2308 . 2 (∃𝑦(𝐴𝑦𝑦𝐵) ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
131, 12bitri 264 1 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1744  wcel 2030  wnfc 2780   cuni 4468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-uni 4469
This theorem is referenced by:  eluni2f  39600  stoweidlem46  40581  stoweidlem57  40592
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