Theorem nom55 336

Description: Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. (Contributed by NM, 7-Feb-1999.)

Assertion

Ref	Expression
nom55	((a ∪ b) ≡ b) = (a →2 b)

Proof of Theorem nom55

Step	Hyp	Ref	Expression
1		nom25 318	. 2 (b⊥ ≡ (b⊥ ∩ a⊥ )) = (b⊥ →1 a⊥ )
2		conb 122	. . 3 ((a ∪ b) ≡ b) = ((a ∪ b)⊥ ≡ b⊥ )
3		bicom 96	. . 3 ((a ∪ b)⊥ ≡ b⊥ ) = (b⊥ ≡ (a ∪ b)⊥ )
4		ancom 74	. . . . . 6 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
5		anor3 90	. . . . . 6 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
6	4, 5	ax-r2 36	. . . . 5 (b⊥ ∩ a⊥ ) = (a ∪ b)⊥
7	6	ax-r1 35	. . . 4 (a ∪ b)⊥ = (b⊥ ∩ a⊥ )
8	7	lbi 97	. . 3 (b⊥ ≡ (a ∪ b)⊥ ) = (b⊥ ≡ (b⊥ ∩ a⊥ ))
9	2, 3, 8	3tr 65	. 2 ((a ∪ b) ≡ b) = (b⊥ ≡ (b⊥ ∩ a⊥ ))
10		i2i1 267	. 2 (a →2 b) = (b⊥ →1 a⊥ )
11	1, 9, 10	3tr1 63	1 ((a ∪ b) ≡ b) = (a →2 b)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45

This theorem is used by:  nom65  342