Theorem soeq2 5555

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

Assertion

Ref	Expression
soeq2	⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵))

Proof of Theorem soeq2

Step	Hyp	Ref	Expression
1		soss 5553	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴))
2		soss 5553	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Or 𝐴 → 𝑅 Or 𝐵))
3	1, 2	anim12i 619	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → ((𝑅 Or 𝐵 → 𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴 → 𝑅 Or 𝐵)))
4		eqss 3937	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
5		dfbi2 475	. . 3 ⊢ ((𝑅 Or 𝐵 ↔ 𝑅 Or 𝐴) ↔ ((𝑅 Or 𝐵 → 𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴 → 𝑅 Or 𝐵)))
6	3, 4, 5	3imtr4i 293	. 2 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐵 ↔ 𝑅 Or 𝐴))
7	6	bicomd 224	1 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ⊆ wss 3890   Or wor 5532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2732  df-ral 3055  df-ss 3907  df-po 5533  df-so 5534

This theorem is referenced by:  soeq12d  5556  weeq2  5613  wemapso2  9465  oemapso  9601  fin2i  10215  isfin2-2  10239  fin1a2lem10  10329  zorn2lem7  10422  zornn0g  10425  opsrtoslem2  22039  ltssolem1  27664  aomclem1  43506