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Theorem elpreq 30301
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 488 . . 3 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝐴)
2 elpreq.3 . . . 4 (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))
32biimpa 480 . . 3 ((𝜑𝑋 = 𝐴) → 𝑌 = 𝐴)
41, 3eqtr4d 2862 . 2 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝑌)
5 elpreq.1 . . . . 5 (𝜑𝑋 ∈ {𝐴, 𝐵})
6 elpri 4572 . . . . 5 (𝑋 ∈ {𝐴, 𝐵} → (𝑋 = 𝐴𝑋 = 𝐵))
75, 6syl 17 . . . 4 (𝜑 → (𝑋 = 𝐴𝑋 = 𝐵))
87orcanai 1000 . . 3 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝐵)
9 simpl 486 . . . 4 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝜑)
102notbid 321 . . . . 5 (𝜑 → (¬ 𝑋 = 𝐴 ↔ ¬ 𝑌 = 𝐴))
1110biimpa 480 . . . 4 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → ¬ 𝑌 = 𝐴)
12 elpreq.2 . . . . 5 (𝜑𝑌 ∈ {𝐴, 𝐵})
13 elpri 4572 . . . . 5 (𝑌 ∈ {𝐴, 𝐵} → (𝑌 = 𝐴𝑌 = 𝐵))
14 pm2.53 848 . . . . 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 2862 . 2 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝑌)
184, 17pm2.61dan 812 1 (𝜑𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  {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-v 3482  df-un 3924  df-sn 4551  df-pr 4553
This theorem is referenced by:  indpreima  31341
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