Theorem elpreq 28744

Description: Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)

Hypotheses

Ref	Expression
elpreq.1	⊢ (𝜑 → 𝑋 ∈ {𝐴, 𝐵})
elpreq.2	⊢ (𝜑 → 𝑌 ∈ {𝐴, 𝐵})
elpreq.3	⊢ (𝜑 → (𝑋 = 𝐴 ↔ 𝑌 = 𝐴))

Assertion

Ref	Expression
elpreq	⊢ (𝜑 → 𝑋 = 𝑌)

Proof of Theorem elpreq

Step	Hyp	Ref	Expression
1		simpr 476	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐴) → 𝑋 = 𝐴)
2		elpreq.3	. . . 4 ⊢ (𝜑 → (𝑋 = 𝐴 ↔ 𝑌 = 𝐴))
3	2	biimpa 500	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐴) → 𝑌 = 𝐴)
4	1, 3	eqtr4d 2647	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝐴) → 𝑋 = 𝑌)
5		elpreq.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ {𝐴, 𝐵})
6		elpri 4145	. . . . 5 ⊢ (𝑋 ∈ {𝐴, 𝐵} → (𝑋 = 𝐴 ∨ 𝑋 = 𝐵))
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 = 𝐴 ∨ 𝑋 = 𝐵))
8	7	orcanai 950	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝐵)
9		simpl 472	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝜑)
10	2	notbid 307	. . . . 5 ⊢ (𝜑 → (¬ 𝑋 = 𝐴 ↔ ¬ 𝑌 = 𝐴))
11	10	biimpa 500	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐴) → ¬ 𝑌 = 𝐴)
12		elpreq.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ {𝐴, 𝐵})
13		elpri 4145	. . . . 5 ⊢ (𝑌 ∈ {𝐴, 𝐵} → (𝑌 = 𝐴 ∨ 𝑌 = 𝐵))
14		pm2.53 387	. . . . 5 ⊢ ((𝑌 = 𝐴 ∨ 𝑌 = 𝐵) → (¬ 𝑌 = 𝐴 → 𝑌 = 𝐵))
15	12, 13, 14	3syl 18	. . . 4 ⊢ (𝜑 → (¬ 𝑌 = 𝐴 → 𝑌 = 𝐵))
16	9, 11, 15	sylc 63	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑌 = 𝐵)
17	8, 16	eqtr4d 2647	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝐴) → 𝑋 = 𝑌)
18	4, 17	pm2.61dan 828	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cpr 4127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128

This theorem is referenced by:  indpreima  29414