Theorem oa4dcom 970

Description: Lemma commuting terms. (Contributed by NM, 24-Dec-1998.)

Assertion

Ref	Expression
oa4dcom	(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)))))) = (b ∩ (((b ∩ a) ∪ ((b →1 d) ∩ (a →1 d))) ∪ (((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))))))

Proof of Theorem oa4dcom

Step	Hyp	Ref	Expression
1		ancom 74	. . . 4 (a ∩ b) = (b ∩ a)
2		ancom 74	. . . 4 ((a →1 d) ∩ (b →1 d)) = ((b →1 d) ∩ (a →1 d))
3	1, 2	2or 72	. . 3 ((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) = ((b ∩ a) ∪ ((b →1 d) ∩ (a →1 d)))
4		ancom 74	. . 3 (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))) = (((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))))
5	3, 4	2or 72	. 2 (((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 ∩ a) ∪ ((b →1 d) ∩ (a →1 d))) ∪ (((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))))
6	5	lan 77	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)))))) = (b ∩ (((b ∩ a) ∪ ((b →1 d) ∩ (a →1 d))) ∪ (((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))))))

Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This theorem was proved from axioms:  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40

This theorem is referenced by:  axoa4d  1038