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Theorem elpreq 29927
 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 479 . . 3 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝐴)
2 elpreq.3 . . . 4 (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))
32biimpa 470 . . 3 ((𝜑𝑋 = 𝐴) → 𝑌 = 𝐴)
41, 3eqtr4d 2817 . 2 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝑌)
5 elpreq.1 . . . . 5 (𝜑𝑋 ∈ {𝐴, 𝐵})
6 elpri 4420 . . . . 5 (𝑋 ∈ {𝐴, 𝐵} → (𝑋 = 𝐴𝑋 = 𝐵))
75, 6syl 17 . . . 4 (𝜑 → (𝑋 = 𝐴𝑋 = 𝐵))
87orcanai 988 . . 3 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝐵)
9 simpl 476 . . . 4 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝜑)
102notbid 310 . . . . 5 (𝜑 → (¬ 𝑋 = 𝐴 ↔ ¬ 𝑌 = 𝐴))
1110biimpa 470 . . . 4 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → ¬ 𝑌 = 𝐴)
12 elpreq.2 . . . . 5 (𝜑𝑌 ∈ {𝐴, 𝐵})
13 elpri 4420 . . . . 5 (𝑌 ∈ {𝐴, 𝐵} → (𝑌 = 𝐴𝑌 = 𝐵))
14 pm2.53 840 . . . . 5 ((𝑌 = 𝐴𝑌 = 𝐵) → (¬ 𝑌 = 𝐴𝑌 = 𝐵))
1512, 13, 143syl 18 . . . 4 (𝜑 → (¬ 𝑌 = 𝐴𝑌 = 𝐵))
169, 11, 15sylc 65 . . 3 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑌 = 𝐵)
178, 16eqtr4d 2817 . 2 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝑌)
184, 17pm2.61dan 803 1 (𝜑𝑋 = 𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2107  {cpr 4400 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-un 3797  df-sn 4399  df-pr 4401 This theorem is referenced by:  indpreima  30689
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