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Theorem eupth2lem2 30068
Description: Lemma for eupth2 30088. (Contributed by Mario Carneiro, 8-Apr-2015.)
Hypothesis
Ref Expression
eupth2lem2.1 𝐵 ∈ V
Assertion
Ref Expression
eupth2lem2 ((𝐵𝐶𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))

Proof of Theorem eupth2lem2
StepHypRef Expression
1 eqidd 2726 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → 𝐵 = 𝐵)
21olcd 872 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴𝐵 = 𝐵))
32biantrud 530 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵 ↔ (𝐴𝐵 ∧ (𝐵 = 𝐴𝐵 = 𝐵))))
4 eupth2lem2.1 . . . . . 6 𝐵 ∈ V
5 eupth2lem1 30067 . . . . . 6 (𝐵 ∈ V → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝐵 = 𝐴𝐵 = 𝐵))))
64, 5ax-mp 5 . . . . 5 (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝐵 = 𝐴𝐵 = 𝐵)))
73, 6bitr4di 288 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
8 simpr 483 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → 𝐵 = 𝑈)
98eleq1d 2810 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
107, 9bitrd 278 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
1110necon1bbid 2970 . 2 ((𝐵𝐶𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝐴 = 𝐵))
12 simpl 481 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → 𝐵𝐶)
13 neeq1 2993 . . . . . . 7 (𝐵 = 𝐴 → (𝐵𝐶𝐴𝐶))
1412, 13syl5ibcom 244 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴𝐴𝐶))
1514pm4.71rd 561 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐴𝐶𝐵 = 𝐴)))
16 eqcom 2732 . . . . 5 (𝐴 = 𝐵𝐵 = 𝐴)
17 ancom 459 . . . . 5 ((𝐵 = 𝐴𝐴𝐶) ↔ (𝐴𝐶𝐵 = 𝐴))
1815, 16, 173bitr4g 313 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐵 = 𝐴𝐴𝐶)))
1912neneqd 2935 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → ¬ 𝐵 = 𝐶)
20 biorf 934 . . . . . . 7 𝐵 = 𝐶 → (𝐵 = 𝐴 ↔ (𝐵 = 𝐶𝐵 = 𝐴)))
2119, 20syl 17 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐵 = 𝐶𝐵 = 𝐴)))
22 orcom 868 . . . . . 6 ((𝐵 = 𝐶𝐵 = 𝐴) ↔ (𝐵 = 𝐴𝐵 = 𝐶))
2321, 22bitrdi 286 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
2423anbi1d 629 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → ((𝐵 = 𝐴𝐴𝐶) ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶)))
2518, 24bitrd 278 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶)))
26 ancom 459 . . 3 ((𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶)) ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶))
2725, 26bitr4di 288 . 2 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶))))
28 eupth2lem1 30067 . . . 4 (𝐵 ∈ V → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ (𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶))))
294, 28ax-mp 5 . . 3 (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ (𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶)))
308eleq1d 2810 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
3129, 30bitr3id 284 . 2 ((𝐵𝐶𝐵 = 𝑈) → ((𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶)) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
3211, 27, 313bitrd 304 1 ((𝐵𝐶𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2930  Vcvv 3463  c0 4319  ifcif 4525  {cpr 4627
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 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-v 3465  df-dif 3944  df-un 3946  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628
This theorem is referenced by:  eupth2lem3lem4  30080
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