Theorem oa4dtoc 949

Description: Derivation of 4-OA law variant.

Hypothesis

Ref	Expression
oa4dtoc.1	(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)

Assertion

Ref	Expression
oa4dtoc	(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 oa4dtoc

Step	Hyp	Ref	Expression
1		oa4dtoc.1	. 2 (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)
2	1	oatr 908	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 was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  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

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