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Theorem eupth2lem1 27341
Description: Lemma for eupth2 27362. (Contributed by Mario Carneiro, 8-Apr-2015.)
Assertion
Ref Expression
eupth2lem1 (𝑈𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))

Proof of Theorem eupth2lem1
StepHypRef Expression
1 eleq2 2816 . . 3 (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ ∅ ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
21bibi1d 332 . 2 (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → ((𝑈 ∈ ∅ ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))) ↔ (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))))
3 eleq2 2816 . . 3 ({𝐴, 𝐵} = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ {𝐴, 𝐵} ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
43bibi1d 332 . 2 ({𝐴, 𝐵} = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → ((𝑈 ∈ {𝐴, 𝐵} ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))) ↔ (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))))
5 noel 4050 . . . 4 ¬ 𝑈 ∈ ∅
65a1i 11 . . 3 ((𝑈𝑉𝐴 = 𝐵) → ¬ 𝑈 ∈ ∅)
7 simpl 474 . . . . 5 ((𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)) → 𝐴𝐵)
87neneqd 2925 . . . 4 ((𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)) → ¬ 𝐴 = 𝐵)
9 simpr 479 . . . 4 ((𝑈𝑉𝐴 = 𝐵) → 𝐴 = 𝐵)
108, 9nsyl3 133 . . 3 ((𝑈𝑉𝐴 = 𝐵) → ¬ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))
116, 102falsed 365 . 2 ((𝑈𝑉𝐴 = 𝐵) → (𝑈 ∈ ∅ ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
12 elprg 4329 . . 3 (𝑈𝑉 → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝑈 = 𝐴𝑈 = 𝐵)))
13 df-ne 2921 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
14 ibar 526 . . . 4 (𝐴𝐵 → ((𝑈 = 𝐴𝑈 = 𝐵) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
1513, 14sylbir 225 . . 3 𝐴 = 𝐵 → ((𝑈 = 𝐴𝑈 = 𝐵) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
1612, 15sylan9bb 738 . 2 ((𝑈𝑉 ∧ ¬ 𝐴 = 𝐵) → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
172, 4, 11, 16ifbothda 4255 1 (𝑈𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1620  wcel 2127  wne 2920  c0 4046  ifcif 4218  {cpr 4311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-v 3330  df-dif 3706  df-un 3708  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312
This theorem is referenced by:  eupth2lem2  27342  eupth2lem3lem6  27356
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