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Theorem eldifpr 4175
Description: Membership in a set with two elements removed. Similar to eldifsn 4287 and eldiftp 4199. (Contributed by Mario Carneiro, 18-Jul-2017.)
Assertion
Ref Expression
eldifpr (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))

Proof of Theorem eldifpr
StepHypRef Expression
1 elprg 4167 . . . . 5 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷)))
21notbid 308 . . . 4 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷)))
3 neanior 2882 . . . 4 ((𝐴𝐶𝐴𝐷) ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷))
42, 3syl6bbr 278 . . 3 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴𝐶𝐴𝐷)))
54pm5.32i 668 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷)))
6 eldif 3565 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷}))
7 3anass 1040 . 2 ((𝐴𝐵𝐴𝐶𝐴𝐷) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷)))
85, 6, 73bitr4i 292 1 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  cdif 3552  {cpr 4150
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-v 3188  df-dif 3558  df-un 3560  df-sn 4149  df-pr 4151
This theorem is referenced by:  rexdifpr  4176  logbcl  24405  logbid1  24406  logb1  24407  elogb  24408  logbchbase  24409  relogbval  24410  relogbcl  24411  relogbreexp  24413  relogbmul  24415  relogbexp  24418  nnlogbexp  24419  relogbcxp  24423  cxplogb  24424  relogbcxpb  24425  logbmpt  24426  logbfval  24428  eluz2cnn0n1  41589  rege1logbrege0  41644  relogbmulbexp  41647  relogbdivb  41648  nnpw2blen  41666
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