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Theorem rexdifpr 4183
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 4182 . . . . 5 (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥𝐴𝑥𝐵𝑥𝐶))
2 3anass 1040 . . . . 5 ((𝑥𝐴𝑥𝐵𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
31, 2bitri 264 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
43anbi1i 730 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑))
5 anass 680 . . . 4 (((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑) ↔ (𝑥𝐴 ∧ ((𝑥𝐵𝑥𝐶) ∧ 𝜑)))
6 df-3an 1038 . . . . . 6 ((𝑥𝐵𝑥𝐶𝜑) ↔ ((𝑥𝐵𝑥𝐶) ∧ 𝜑))
76bicomi 214 . . . . 5 (((𝑥𝐵𝑥𝐶) ∧ 𝜑) ↔ (𝑥𝐵𝑥𝐶𝜑))
87anbi2i 729 . . . 4 ((𝑥𝐴 ∧ ((𝑥𝐵𝑥𝐶) ∧ 𝜑)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
95, 8bitri 264 . . 3 (((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
104, 9bitri 264 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
1110rexbii2 3034 1 (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036  wcel 1987  wne 2790  wrex 2909  cdif 3557  {cpr 4157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rex 2914  df-v 3192  df-dif 3563  df-un 3565  df-sn 4156  df-pr 4158
This theorem is referenced by:  usgr2pth0  26564
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