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Theorem elpreq 28532
Description: Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
Hypotheses
Ref Expression
elpreq.1 (𝜑𝑋 ∈ {𝐴, 𝐵})
elpreq.2 (𝜑𝑌 ∈ {𝐴, 𝐵})
elpreq.3 (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))
Assertion
Ref Expression
elpreq (𝜑𝑋 = 𝑌)

Proof of Theorem elpreq
StepHypRef Expression
1 simpr 475 . . 3 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝐴)
2 elpreq.3 . . . 4 (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))
32biimpa 499 . . 3 ((𝜑𝑋 = 𝐴) → 𝑌 = 𝐴)
41, 3eqtr4d 2551 . 2 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝑌)
5 elpreq.1 . . . . 5 (𝜑𝑋 ∈ {𝐴, 𝐵})
6 elpri 4048 . . . . 5 (𝑋 ∈ {𝐴, 𝐵} → (𝑋 = 𝐴𝑋 = 𝐵))
75, 6syl 17 . . . 4 (𝜑 → (𝑋 = 𝐴𝑋 = 𝐵))
87orcanai 949 . . 3 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝐵)
9 simpl 471 . . . 4 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝜑)
102notbid 306 . . . . 5 (𝜑 → (¬ 𝑋 = 𝐴 ↔ ¬ 𝑌 = 𝐴))
1110biimpa 499 . . . 4 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → ¬ 𝑌 = 𝐴)
12 elpreq.2 . . . . 5 (𝜑𝑌 ∈ {𝐴, 𝐵})
13 elpri 4048 . . . . 5 (𝑌 ∈ {𝐴, 𝐵} → (𝑌 = 𝐴𝑌 = 𝐵))
14 pm2.53 386 . . . . 5 ((𝑌 = 𝐴𝑌 = 𝐵) → (¬ 𝑌 = 𝐴𝑌 = 𝐵))
1512, 13, 143syl 18 . . . 4 (𝜑 → (¬ 𝑌 = 𝐴𝑌 = 𝐵))
169, 11, 15sylc 62 . . 3 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑌 = 𝐵)
178, 16eqtr4d 2551 . 2 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝑌)
184, 17pm2.61dan 827 1 (𝜑𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1938  {cpr 4030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-v 3079  df-un 3449  df-sn 4029  df-pr 4031
This theorem is referenced by:  indpreima  29210
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