Theorem axoa4b 1014

Description: Proper 4-variable OA law variant.

Assertion

Ref	Expression
axoa4b	((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

Proof of Theorem axoa4b

Step	Hyp	Ref	Expression
1		axoa4 1013	. 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)))))))) ≤ d
2	1	oa4ctob 947	1 ((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

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

This theorem is referenced by:  oa6 1015

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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