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Theorem eupth2lem2 27399
Description: Lemma for eupth2 27419. (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 2772 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → 𝐵 = 𝐵)
21olcd 861 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴𝐵 = 𝐵))
32biantrud 521 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵 ↔ (𝐴𝐵 ∧ (𝐵 = 𝐴𝐵 = 𝐵))))
4 eupth2lem2.1 . . . . . 6 𝐵 ∈ V
5 eupth2lem1 27398 . . . . . 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, 6syl6bbr 278 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
8 simpr 471 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → 𝐵 = 𝑈)
98eleq1d 2835 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
107, 9bitrd 268 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
1110necon1bbid 2982 . 2 ((𝐵𝐶𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝐴 = 𝐵))
12 simpl 468 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → 𝐵𝐶)
13 neeq1 3005 . . . . . . 7 (𝐵 = 𝐴 → (𝐵𝐶𝐴𝐶))
1412, 13syl5ibcom 235 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴𝐴𝐶))
1514pm4.71rd 552 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐴𝐶𝐵 = 𝐴)))
16 eqcom 2778 . . . . 5 (𝐴 = 𝐵𝐵 = 𝐴)
17 ancom 452 . . . . 5 ((𝐵 = 𝐴𝐴𝐶) ↔ (𝐴𝐶𝐵 = 𝐴))
1815, 16, 173bitr4g 303 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐵 = 𝐴𝐴𝐶)))
1912neneqd 2948 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → ¬ 𝐵 = 𝐶)
20 biorf 920 . . . . . . 7 𝐵 = 𝐶 → (𝐵 = 𝐴 ↔ (𝐵 = 𝐶𝐵 = 𝐴)))
2119, 20syl 17 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐵 = 𝐶𝐵 = 𝐴)))
22 orcom 857 . . . . . 6 ((𝐵 = 𝐶𝐵 = 𝐴) ↔ (𝐵 = 𝐴𝐵 = 𝐶))
2321, 22syl6bb 276 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
2423anbi1d 615 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → ((𝐵 = 𝐴𝐴𝐶) ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶)))
2518, 24bitrd 268 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶)))
26 ancom 452 . . 3 ((𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶)) ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶))
2725, 26syl6bbr 278 . 2 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶))))
28 eupth2lem1 27398 . . . 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 2835 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
3129, 30syl5bbr 274 . 2 ((𝐵𝐶𝐵 = 𝑈) → ((𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶)) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
3211, 27, 313bitrd 294 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 196  wa 382  wo 834   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  c0 4063  ifcif 4225  {cpr 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319
This theorem is referenced by:  eupth2lem3lem4  27411
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