Theorem axoa4 1013

Description: The proper 4-variable OA law.

Assertion

Ref	Expression
axoa4	(a⊥ ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))))) ≤ d

Proof of Theorem axoa4

Step	Hyp	Ref	Expression
1		u1lem9b 760	. . 3 a⊥ ≤ (a →1 d)
2	1	leran 145	. 2 (a⊥ ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))))) ≤ ((a →1 d) ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))))))
3		ax-4oa 1012	. . . 4 (((b →1 d) →1 d) ∩ ((((b →1 d) ∩ (a →1 d)) ∪ (((b →1 d) →1 d) ∩ ((a →1 d) →1 d))) ∪ ((((b →1 d) ∩ (c →1 d)) ∪ (((b →1 d) →1 d) ∩ ((c →1 d) →1 d))) ∩ (((a →1 d) ∩ (c →1 d)) ∪ (((a →1 d) →1 d) ∩ ((c →1 d) →1 d)))))) ≤ ((a →1 d) →1 d)
4		id 58	. . . 4 (a →1 d) = (a →1 d)
5		id 58	. . . 4 (b →1 d) = (b →1 d)
6		id 58	. . . 4 (c →1 d) = (c →1 d)
7	3, 4, 5, 6	oa4gto4u 956	. . 3 ((a →1 d) ∩ ((a⊥ →1 d) ∪ ((b⊥ →1 d) ∩ ((((a →1 d) ∩ (b →1 d)) ∪ ((a⊥ →1 d) ∩ (b⊥ →1 d))) ∪ ((((a →1 d) ∩ (c →1 d)) ∪ ((a⊥ →1 d) ∩ (c⊥ →1 d))) ∩ (((b →1 d) ∩ (c →1 d)) ∪ ((b⊥ →1 d) ∩ (c⊥ →1 d)))))))) ≤ d
8	7	oa4uto4 957	. 2 ((a →1 d) ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))))) ≤ d
9	2, 8	letr 129	1 (a⊥ ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))))) ≤ d

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

This theorem is referenced by:  axoa4b 1014  axoa4d 1017

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-4oa 1012

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

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