3oa2 - Quantum Logic Explorer

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Theorem 3oa2 1024

Description: Alternate form for the 3-variable orthoarguesion law. (Contributed by NM, 27-May-2004.)

**Assertion**
Ref	Expression
3oa2	((a →₁c) ∩ (((a →₁c) ∩ (b →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c)))) ≤ (b →₁c)

**Proof of Theorem 3oa2**
Step	Hyp	Ref	Expression
1		ax-3oa 998	. 2 (((a^⊥ →₁c) →₁c) ∩ (((a^⊥ →₁c) ∩ (b^⊥ →₁c)) ∪ (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)))) ≤ ((b^⊥ →₁c) →₁c)
2		u1lem11 780	. . 3 ((a^⊥ →₁c) →₁c) = (a →₁c)
3		ax-a2 31	. . . 4 (((a^⊥ →₁c) ∩ (b^⊥ →₁c)) ∪ (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c))) = ((((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c)))
4		u1lem11 780	. . . . . 6 ((b^⊥ →₁c) →₁c) = (b →₁c)
5	2, 4	2an 79	. . . . 5 (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)) = ((a →₁c) ∩ (b →₁c))
6	5	ax-r5 38	. . . 4 ((((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c))) = (((a →₁c) ∩ (b →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c)))
7	3, 6	ax-r2 36	. . 3 (((a^⊥ →₁c) ∩ (b^⊥ →₁c)) ∪ (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c))) = (((a →₁c) ∩ (b →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c)))
8	2, 7	2an 79	. 2 (((a^⊥ →₁c) →₁c) ∩ (((a^⊥ →₁c) ∩ (b^⊥ →₁c)) ∪ (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)))) = ((a →₁c) ∩ (((a →₁c) ∩ (b →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c))))
9	1, 8, 4	le3tr2 141	1 ((a →₁c) ∩ (((a →₁c) ∩ (b →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c)))) ≤ (b →₁c)

Colors of variables: term

Syntax hints: ≤ wle 2 ^⊥ wn 4 ∪ wo 6 ∩ wa 7 →₁wi1 12

This theorem was proved from axioms: ax-a1 30 ax-a2 31 ax-a3 32 ax-a4 33 ax-a5 34 ax-r1 35 ax-r2 36 ax-r4 37 ax-r5 38 ax-r3 439 ax-3oa 998

This theorem depends on definitions: df-b 39 df-a 40 df-t 41 df-f 42 df-i1 44 df-le1 130 df-le2 131 df-c1 132 df-c2 133

This theorem is referenced by: 3oa3 1025