Theorem i3orlem6 557

Description: Lemma for Kalmbach implication OR builder. (Contributed by NM, 11-Nov-1997.)

Assertion

Ref	Expression
i3orlem6	((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c))) = (((a ∪ b) ∩ (a⊥ →3 b⊥ )) ∪ ((a ∪ c) →3 (b ∪ c)))

Proof of Theorem i3orlem6

Step	Hyp	Ref	Expression
1		ax-a3 32	. . 3 (((a ∩ b) ∪ (a →3 b)⊥ ) ∪ ((a ∪ c) →3 (b ∪ c))) = ((a ∩ b) ∪ ((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c))))
2	1	ax-r1 35	. 2 ((a ∩ b) ∪ ((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))) = (((a ∩ b) ∪ (a →3 b)⊥ ) ∪ ((a ∪ c) →3 (b ∪ c)))
3		i3orlem2 553	. . . 4 (a ∩ b) ≤ ((a ∪ c) →3 (b ∪ c))
4	3	lerr 150	. . 3 (a ∩ b) ≤ ((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))
5	4	df-le2 131	. 2 ((a ∩ b) ∪ ((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))) = ((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))
6		oi3ai3 503	. . 3 ((a ∩ b) ∪ (a →3 b)⊥ ) = ((a ∪ b) ∩ (a⊥ →3 b⊥ ))
7	6	ax-r5 38	. 2 (((a ∩ b) ∪ (a →3 b)⊥ ) ∪ ((a ∪ c) →3 (b ∪ c))) = (((a ∪ b) ∩ (a⊥ →3 b⊥ )) ∪ ((a ∪ c) →3 (b ∪ c)))
8	2, 5, 7	3tr2 64	1 ((a →3 b)⊥ ∪ ((a ∪ c) →3 (b ∪ c))) = (((a ∪ b) ∩ (a⊥ →3 b⊥ )) ∪ ((a ∪ c) →3 (b ∪ c)))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  i3orlem7  558  i3orlem8  559