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Theorem elpreq 32456
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 483 . . 3 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝐴)
2 elpreq.3 . . . 4 (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))
32biimpa 475 . . 3 ((𝜑𝑋 = 𝐴) → 𝑌 = 𝐴)
41, 3eqtr4d 2769 . 2 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝑌)
5 elpreq.1 . . . . 5 (𝜑𝑋 ∈ {𝐴, 𝐵})
6 elpri 4656 . . . . 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 481 . . . 4 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝜑)
102notbid 317 . . . . 5 (𝜑 → (¬ 𝑋 = 𝐴 ↔ ¬ 𝑌 = 𝐴))
1110biimpa 475 . . . 4 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → ¬ 𝑌 = 𝐴)
12 elpreq.2 . . . . 5 (𝜑𝑌 ∈ {𝐴, 𝐵})
13 elpri 4656 . . . . 5 (𝑌 ∈ {𝐴, 𝐵} → (𝑌 = 𝐴𝑌 = 𝐵))
14 pm2.53 849 . . . . 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 2769 . 2 ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝑌)
184, 17pm2.61dan 811 1 (𝜑𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1534  wcel 2099  {cpr 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-un 3952  df-sn 4634  df-pr 4636
This theorem is referenced by:  indpreima  33858
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