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Theorem rexdifpr 4426
Description: Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
Assertion
Ref Expression
rexdifpr (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))

Proof of Theorem rexdifpr
StepHypRef Expression
1 eldifpr 4425 . . . . 5 (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥𝐴𝑥𝐵𝑥𝐶))
2 3anass 1079 . . . . 5 ((𝑥𝐴𝑥𝐵𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
31, 2bitri 267 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
43anbi1i 617 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑))
5 anass 462 . . . 4 (((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑) ↔ (𝑥𝐴 ∧ ((𝑥𝐵𝑥𝐶) ∧ 𝜑)))
6 df-3an 1073 . . . . . 6 ((𝑥𝐵𝑥𝐶𝜑) ↔ ((𝑥𝐵𝑥𝐶) ∧ 𝜑))
76bicomi 216 . . . . 5 (((𝑥𝐵𝑥𝐶) ∧ 𝜑) ↔ (𝑥𝐵𝑥𝐶𝜑))
87anbi2i 616 . . . 4 ((𝑥𝐴 ∧ ((𝑥𝐵𝑥𝐶) ∧ 𝜑)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
95, 8bitri 267 . . 3 (((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
104, 9bitri 267 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
1110rexbii2 3221 1 (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  w3a 1071  wcel 2106  wne 2968  wrex 3090  cdif 3788  {cpr 4399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-rex 3095  df-v 3399  df-dif 3794  df-un 3796  df-sn 4398  df-pr 4400
This theorem is referenced by:  usgr2pth0  27117
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