Theorem u2lemab 611

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

Assertion

Ref	Expression
u2lemab	((a →2 b) ∩ b) = b

Proof of Theorem u2lemab

Step	Hyp	Ref	Expression
1		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2	1	ran 78	. 2 ((a →2 b) ∩ b) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ b)
3		ancom 74	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∩ b) = (b ∩ (b ∪ (a⊥ ∩ b⊥ )))
4		anabs 121	. . 3 (b ∩ (b ∪ (a⊥ ∩ b⊥ ))) = b
5	3, 4	ax-r2 36	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∩ b) = b
6	2, 5	ax-r2 36	1 ((a →2 b) ∩ b) = b

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-i2 45

This theorem is referenced by:  u2lemnonb  676  u21lembi  727  bi3  839  bi4  840