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Theorem oa3-1to5 973

Description: Derivation of an equivalent of the second "universal" 3-OA U2 from an equivalent of the first "universal" 3-OA U1. This shows that U2 is redundant in a system containg U1. The hypothesis is theorem oal1 980.

**Hypothesis**
Ref	Expression
oa3-1to5.1	((a →₁c) ∩ ((a ∩ b) ∪ ((a →₁c) ∩ (b →₁c)))) ≤ (b →₁c)

**Assertion**
Ref	Expression
oa3-1to5	(c ∩ ((b →₁c) ∪ ((a →₁c) ∩ ((a ∩ b) ∪ ((a →₁c) ∩ (b →₁c)))))) ≤ (b^⊥ →₁c)

**Proof of Theorem oa3-1to5**
Step	Hyp	Ref	Expression
1		leid 140	. . . . 5 (b →₁c) ≤ (b →₁c)
2		oa3-1to5.1	. . . . 5 ((a →₁c) ∩ ((a ∩ b) ∪ ((a →₁c) ∩ (b →₁c)))) ≤ (b →₁c)
3	1, 2	lel2or 162	. . . 4 ((b →₁c) ∪ ((a →₁c) ∩ ((a ∩ b) ∪ ((a →₁c) ∩ (b →₁c))))) ≤ (b →₁c)
4	3	lelan 159	. . 3 (c ∩ ((b →₁c) ∪ ((a →₁c) ∩ ((a ∩ b) ∪ ((a →₁c) ∩ (b →₁c)))))) ≤ (c ∩ (b →₁c))
5		ax-a1 29	. . . . . . . 8 b = b^⊥ ^⊥
6	5	ran 71	. . . . . . 7 (b ∩ c) = (b^⊥ ^⊥ ∩ c)
7	6	ax-r5 37	. . . . . 6 ((b ∩ c) ∪ (b^⊥ ∩ c)) = ((b^⊥ ^⊥ ∩ c) ∪ (b^⊥ ∩ c))
8		ax-a2 30	. . . . . 6 ((b^⊥ ^⊥ ∩ c) ∪ (b^⊥ ∩ c)) = ((b^⊥ ∩ c) ∪ (b^⊥ ^⊥ ∩ c))
9	7, 8	ax-r2 35	. . . . 5 ((b ∩ c) ∪ (b^⊥ ∩ c)) = ((b^⊥ ∩ c) ∪ (b^⊥ ^⊥ ∩ c))
10		u1lemab 592	. . . . 5 ((b →₁c) ∩ c) = ((b ∩ c) ∪ (b^⊥ ∩ c))
11		u1lemab 592	. . . . 5 ((b^⊥ →₁c) ∩ c) = ((b^⊥ ∩ c) ∪ (b^⊥ ^⊥ ∩ c))
12	9, 10, 11	3tr1 60	. . . 4 ((b →₁c) ∩ c) = ((b^⊥ →₁c) ∩ c)
13		ancom 68	. . . 4 (c ∩ (b →₁c)) = ((b →₁c) ∩ c)
14		ancom 68	. . . 4 (c ∩ (b^⊥ →₁c)) = ((b^⊥ →₁c) ∩ c)
15	12, 13, 14	3tr1 60	. . 3 (c ∩ (b →₁c)) = (c ∩ (b^⊥ →₁c))
16	4, 15	lbtr 131	. 2 (c ∩ ((b →₁c) ∪ ((a →₁c) ∩ ((a ∩ b) ∪ ((a →₁c) ∩ (b →₁c)))))) ≤ (c ∩ (b^⊥ →₁c))
17		lear 153	. 2 (c ∩ (b^⊥ →₁c)) ≤ (b^⊥ →₁c)
18	16, 17	letr 129	1 (c ∩ ((b →₁c) ∪ ((a →₁c) ∩ ((a ∩ b) ∪ ((a →₁c) ∩ (b →₁c)))))) ≤ (b^⊥ →₁c)

Colors of variables: term

Syntax hints: ≤ wle 2 ^⊥ wn 4 ∪ wo 6 ∩ wa 7 →₁wi1 13

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

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

metamath.org