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Theorem rexdifpr 4666
Description: Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.) (Proof shortened by Wolf Lammen, 15-May-2025.)
Assertion
Ref Expression
rexdifpr (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))

Proof of Theorem rexdifpr
StepHypRef Expression
1 anass 467 . . 3 (((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑) ↔ (𝑥𝐴 ∧ ((𝑥𝐵𝑥𝐶) ∧ 𝜑)))
2 eldifpr 4665 . . . . 5 (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥𝐴𝑥𝐵𝑥𝐶))
3 3anass 1092 . . . . 5 ((𝑥𝐴𝑥𝐵𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
42, 3bitri 274 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
54anbi1i 622 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑))
6 df-3an 1086 . . . 4 ((𝑥𝐵𝑥𝐶𝜑) ↔ ((𝑥𝐵𝑥𝐶) ∧ 𝜑))
76anbi2i 621 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)) ↔ (𝑥𝐴 ∧ ((𝑥𝐵𝑥𝐶) ∧ 𝜑)))
81, 5, 73bitr4i 302 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
98rexbii2 3087 1 (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  w3a 1084  wcel 2098  wne 2937  wrex 3067  cdif 3946  {cpr 4634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rex 3068  df-v 3475  df-dif 3952  df-un 3954  df-sn 4633  df-pr 4635
This theorem is referenced by:  usgr2pth0  29599
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