Theorem axoa4d 1017

Description: Proper 4-variable OA law variant.

Assertion

Ref	Expression
axoa4d	(a ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))) ≤ (b⊥ →1 d)

Proof of Theorem axoa4d

Step	Hyp	Ref	Expression
1		oa4dcom 950	. . 3 (a ∩ (((b ∩ a) ∪ ((b →1 d) ∩ (a →1 d))) ∪ (((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))))) = (a ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))))
2	1	ax-r1 34	. 2 (a ∩ (((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 ∩ (((b ∩ a) ∪ ((b →1 d) ∩ (a →1 d))) ∪ (((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))))))
3		axoa4 1013	. . 3 (b⊥ ∩ (b ∪ (a ∩ (((b ∩ a) ∪ ((b →1 d) ∩ (a →1 d))) ∪ (((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))))))) ≤ d
4	3	oa4ctod 948	. 2 (a ∩ (((b ∩ a) ∪ ((b →1 d) ∩ (a →1 d))) ∪ (((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))))) ≤ (b⊥ →1 d)
5	2, 4	bltr 130	1 (a ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))) ≤ (b⊥ →1 d)

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