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Theorem eupth2lem2 28007
Description: Lemma for eupth2 28027. (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 2825 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → 𝐵 = 𝐵)
21olcd 871 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴𝐵 = 𝐵))
32biantrud 535 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵 ↔ (𝐴𝐵 ∧ (𝐵 = 𝐴𝐵 = 𝐵))))
4 eupth2lem2.1 . . . . . 6 𝐵 ∈ V
5 eupth2lem1 28006 . . . . . 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 292 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
8 simpr 488 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → 𝐵 = 𝑈)
98eleq1d 2900 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
107, 9bitrd 282 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐴𝐵𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
1110necon1bbid 3053 . 2 ((𝐵𝐶𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝐴 = 𝐵))
12 simpl 486 . . . . . . 7 ((𝐵𝐶𝐵 = 𝑈) → 𝐵𝐶)
13 neeq1 3076 . . . . . . 7 (𝐵 = 𝐴 → (𝐵𝐶𝐴𝐶))
1412, 13syl5ibcom 248 . . . . . 6 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴𝐴𝐶))
1514pm4.71rd 566 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐴𝐶𝐵 = 𝐴)))
16 eqcom 2831 . . . . 5 (𝐴 = 𝐵𝐵 = 𝐴)
17 ancom 464 . . . . 5 ((𝐵 = 𝐴𝐴𝐶) ↔ (𝐴𝐶𝐵 = 𝐴))
1815, 16, 173bitr4g 317 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐵 = 𝐴𝐴𝐶)))
1912neneqd 3019 . . . . . . 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 867 . . . . . 6 ((𝐵 = 𝐶𝐵 = 𝐴) ↔ (𝐵 = 𝐴𝐵 = 𝐶))
2321, 22syl6bb 290 . . . . 5 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
2423anbi1d 632 . . . 4 ((𝐵𝐶𝐵 = 𝑈) → ((𝐵 = 𝐴𝐴𝐶) ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶)))
2518, 24bitrd 282 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶)))
26 ancom 464 . . 3 ((𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶)) ↔ ((𝐵 = 𝐴𝐵 = 𝐶) ∧ 𝐴𝐶))
2725, 26syl6bbr 292 . 2 ((𝐵𝐶𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶))))
28 eupth2lem1 28006 . . . 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 2900 . . 3 ((𝐵𝐶𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
3129, 30bitr3id 288 . 2 ((𝐵𝐶𝐵 = 𝑈) → ((𝐴𝐶 ∧ (𝐵 = 𝐴𝐵 = 𝐶)) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
3211, 27, 313bitrd 308 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 209  wa 399  wo 844   = wceq 1538  wcel 2115  wne 3014  Vcvv 3480  c0 4276  ifcif 4450  {cpr 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3015  df-v 3482  df-dif 3922  df-un 3924  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553
This theorem is referenced by:  eupth2lem3lem4  28019
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