Theorem soeq1 5579

Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)

Assertion

Ref	Expression
soeq1	⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))

Proof of Theorem soeq1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poeq1 5561	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐴))
2		breq 5104	. . . . 5 ⊢ (𝑅 = 𝑆 → (𝑥𝑅𝑦 ↔ 𝑥𝑆𝑦))
3		biidd 265	. . . . 5 ⊢ (𝑅 = 𝑆 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑦))
4		breq 5104	. . . . 5 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑥 ↔ 𝑦𝑆𝑥))
5	2, 3, 4	3orbi123d 1463	. . . 4 ⊢ (𝑅 = 𝑆 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑆𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑆𝑥)))
6	5	2ralbidv 3226	. . 3 ⊢ (𝑅 = 𝑆 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑆𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑆𝑥)))
7	1, 6	anbi12d 644	. 2 ⊢ (𝑅 = 𝑆 → ((𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (𝑆 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑆𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑆𝑥))))
8		df-so 5559	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
9		df-so 5559	. 2 ⊢ (𝑆 Or 𝐴 ↔ (𝑆 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑆𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑆𝑥)))
10	7, 8, 9	3bitr4g 317	1 ⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1569  ∀wral 3076   class class class wbr 5102   Po wpo 5556   Or wor 5557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-ex 1812  df-cleq 2752  df-clel 2835  df-ral 3077  df-br 5103  df-po 5558  df-so 5559

This theorem is used by:  soeq12d  5581  weeq1  5637  ltsopi  10977  cnso  16421