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Theorem eupath2lem2 26505
 Description: Lemma for eupath2 26507. (Contributed by Mario Carneiro, 8-Apr-2015.)
Hypothesis
Ref Expression
eupath2lem2.1 𝐵 ∈ V
Assertion
Ref Expression
eupath2lem2 ((𝐵𝐶𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))

Proof of Theorem eupath2lem2
StepHypRef Expression
1 eqidd 2611 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → 𝐵 = 𝐵)
21olcd 407 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴𝐵 = 𝐵))
32biantrud 527 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵 ↔ (𝐴𝐵 ∧ (𝐵 = 𝐴𝐵 = 𝐵))))
4 eupath2lem2.1 . . . . . 6 𝐵 ∈ V
5 eupath2lem1 26504 . . . . . 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 277 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
8 simpr 476 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → 𝐵 = 𝑈)
98eleq1d 2672 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
107, 9bitrd 267 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
1110necon1bbid 2821 . 2 ((𝐵𝐶𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝐴 = 𝐵))
12 simpl 472 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → 𝐵𝐶)
13 neeq1 2844 . . . . . . 7 (𝐵 = 𝐴 → (𝐵𝐶𝐴𝐶))
1412, 13syl5ibcom 234 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴𝐴𝐶))
1514pm4.71rd 665 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐴𝐶𝐵 = 𝐴)))
16 eqcom 2617 . . . . 5 (𝐴 = 𝐵𝐵 = 𝐴)
17 ancom 465 . . . . 5 ((𝐵 = 𝐴𝐴𝐶) ↔ (𝐴𝐶𝐵 = 𝐴))
1815, 16, 173bitr4g 302 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐵 = 𝐴𝐴𝐶)))
1912neneqd 2787 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → ¬ 𝐵 = 𝐶)
20 biorf 419 . . . . . . 7 𝐵 = 𝐶 → (𝐵 = 𝐴 ↔ (𝐵 = 𝐶𝐵 = 𝐴)))
2119, 20syl 17 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐵 = 𝐶𝐵 = 𝐴)))
22 orcom 401 . . . . . 6 ((𝐵 = 𝐶𝐵 = 𝐴) ↔ (𝐵 = 𝐴𝐵 = 𝐶))
2321, 22syl6bb 275 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
2423anbi1d 737 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → ((𝐵 = 𝐴𝐴𝐶) ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶)))
2518, 24bitrd 267 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶)))
26 ancom 465 . . 3 ((𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶)) ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶))
2725, 26syl6bbr 277 . 2 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶))))
28 eupath2lem1 26504 . . . 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 2672 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
3129, 30syl5bbr 273 . 2 ((𝐵𝐶𝐵 = 𝑈) → ((𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶)) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
3211, 27, 313bitrd 293 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 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  ifcif 4036  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128 This theorem is referenced by:  eupath2lem3  26506  eupth2lem3lem4  41399
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