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Theorem 3oa2 1004

Description: Alternate form for the 3-variable orthoarguesion law.

**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 978	. 2 (((a^⊥ →₁c) →₁c) ∩ (((a^⊥ →₁c) ∩ (b^⊥ →₁c)) ∪ (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)))) ≤ ((b^⊥ →₁c) →₁c)
2		u1lem11 762	. . 3 ((a^⊥ →₁c) →₁c) = (a →₁c)
3		ax-a2 30	. . . 4 (((a^⊥ →₁c) ∩ (b^⊥ →₁c)) ∪ (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c))) = ((((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c)))
4		u1lem11 762	. . . . . 6 ((b^⊥ →₁c) →₁c) = (b →₁c)
5	2, 4	2an 72	. . . . 5 (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)) = ((a →₁c) ∩ (b →₁c))
6	5	ax-r5 37	. . . 4 ((((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c))) = (((a →₁c) ∩ (b →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c)))
7	3, 6	ax-r2 35	. . 3 (((a^⊥ →₁c) ∩ (b^⊥ →₁c)) ∪ (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c))) = (((a →₁c) ∩ (b →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c)))
8	2, 7	2an 72	. 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 133	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 13

This theorem is referenced by: 3oa3 1005

This theorem was proved from axioms: ax-a1 29 ax-a2 30 ax-a3 31 ax-a4 32 ax-a5 33 ax-r1 34 ax-r2 35 ax-r4 36 ax-r5 37 ax-r3 421 ax-3oa 978

This theorem depends on definitions: df-b 38 df-a 39 df-t 40 df-f 41 df-i1 43 df-le1 122 df-le2 123 df-c1 124 df-c2 125