Theorem soeq2 2142

Description: Equality theorem for the strict ordering predicate.

Assertion

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

Proof of Theorem soeq2

Step	Hyp	Ref	Expression
1		soss 2140	. . . 4 ⊢ (A ⊆ B → (R Or B → R Or A))
2		soss 2140	. . . 4 ⊢ (B ⊆ A → (R Or A → R Or B))
3	1, 2	anim12i 268	. . 3 ⊢ ((A ⊆ B ∧ B ⊆ A) → ((R Or B → R Or A) ∧ (R Or A → R Or B)))
4		eqss 1516	. . 3 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		bi 396	. . 3 ⊢ ((R Or B ↔ R Or A) ↔ ((R Or B → R Or A) ∧ (R Or A → R Or B)))
6	3, 4, 5	3imtr4 192	. 2 ⊢ (A = B → (R Or B ↔ R Or A))
7	6	bicomd 399	1 ⊢ (A = B → (R Or A ↔ R Or B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ⊆ wss 1487   Or wor 2059

This theorem is referenced by:  weeq2 2190  zornlem7 3609

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-in 1491  df-ss 1492  df-po 2128  df-so 2138

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