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Theorem elunif 39935
Description: A version of eluni 4631 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 4631 . 2 (𝐴 𝐵 ↔ ∃𝑦(𝐴𝑦𝑦𝐵))
2 elunif.1 . . . . 5 𝑥𝐴
3 nfcv 2941 . . . . 5 𝑥𝑦
42, 3nfel 2954 . . . 4 𝑥 𝐴𝑦
5 elunif.2 . . . . 5 𝑥𝐵
63, 5nfel 2954 . . . 4 𝑥 𝑦𝐵
74, 6nfan 1999 . . 3 𝑥(𝐴𝑦𝑦𝐵)
8 nfv 2010 . . 3 𝑦(𝐴𝑥𝑥𝐵)
9 eleq2w 2862 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
10 eleq1w 2861 . . . 4 (𝑦 = 𝑥 → (𝑦𝐵𝑥𝐵))
119, 10anbi12d 625 . . 3 (𝑦 = 𝑥 → ((𝐴𝑦𝑦𝐵) ↔ (𝐴𝑥𝑥𝐵)))
127, 8, 11cbvexv1 2358 . 2 (∃𝑦(𝐴𝑦𝑦𝐵) ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
131, 12bitri 267 1 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  wex 1875  wcel 2157  wnfc 2928   cuni 4628
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-ext 2777
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 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-uni 4629
This theorem is referenced by:  eluni2f  40044  stoweidlem46  41006  stoweidlem57  41017
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