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Theorem eqoreldifOLD 4197
Description: Obsolete proof of eqoreldif 4196 as of 23-Jul-2021. (Contributed by AV, 25-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
eqoreldifOLD (𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))

Proof of Theorem eqoreldifOLD
StepHypRef Expression
1 orc 400 . . . . 5 (𝐴 = 𝐵 → (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})))
21a1d 25 . . . 4 (𝐴 = 𝐵 → ((𝐵𝐶𝐴𝐶) → (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))
3 simprr 795 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐵𝐶𝐴𝐶)) → 𝐴𝐶)
4 elsni 4165 . . . . . . . . . 10 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
54a1i 11 . . . . . . . . 9 ((𝐵𝐶𝐴𝐶) → (𝐴 ∈ {𝐵} → 𝐴 = 𝐵))
65con3d 148 . . . . . . . 8 ((𝐵𝐶𝐴𝐶) → (¬ 𝐴 = 𝐵 → ¬ 𝐴 ∈ {𝐵}))
76impcom 446 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐵𝐶𝐴𝐶)) → ¬ 𝐴 ∈ {𝐵})
83, 7eldifd 3566 . . . . . 6 ((¬ 𝐴 = 𝐵 ∧ (𝐵𝐶𝐴𝐶)) → 𝐴 ∈ (𝐶 ∖ {𝐵}))
98olcd 408 . . . . 5 ((¬ 𝐴 = 𝐵 ∧ (𝐵𝐶𝐴𝐶)) → (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})))
109ex 450 . . . 4 𝐴 = 𝐵 → ((𝐵𝐶𝐴𝐶) → (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))
112, 10pm2.61i 176 . . 3 ((𝐵𝐶𝐴𝐶) → (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})))
1211ex 450 . 2 (𝐵𝐶 → (𝐴𝐶 → (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))
13 eleq1 2686 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
1413biimprd 238 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
15 eldifi 3710 . . . . 5 (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴𝐶)
1615a1d 25 . . . 4 (𝐴 ∈ (𝐶 ∖ {𝐵}) → (𝐵𝐶𝐴𝐶))
1714, 16jaoi 394 . . 3 ((𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})) → (𝐵𝐶𝐴𝐶))
1817com12 32 . 2 (𝐵𝐶 → ((𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})) → 𝐴𝐶))
1912, 18impbid 202 1 (𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  cdif 3552  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-sn 4149
This theorem is referenced by: (None)
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