Theorem u3lem1 736

Description: Lemma for unified implication study. (Contributed by NM, 14-Dec-1997.)

Assertion

Ref	Expression
u3lem1	((a →3 b) →3 a) = ((a ∪ b) ∩ (a ∪ b⊥ ))

Proof of Theorem u3lem1

Step	Hyp	Ref	Expression
1		comi31 508	. . . 4 a C (a →3 b)
2	1	comcom 453	. . 3 (a →3 b) C a
3	2	u3lemc4 703	. 2 ((a →3 b) →3 a) = ((a →3 b)⊥ ∪ a)
4		u3lemnoa 662	. 2 ((a →3 b)⊥ ∪ a) = ((a ∪ b) ∩ (a ∪ b⊥ ))
5	3, 4	ax-r2 36	1 ((a →3 b) →3 a) = ((a ∪ b) ∩ (a ∪ b⊥ ))

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:  u3lem1n  741