Theorem u2lemab 593

Description: Lemma for Dishkant implication study.

Assertion

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

Proof of Theorem u2lemab

Step	Hyp	Ref	Expression
1		df-i2 44	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2	1	ran 71	. 2 ((a →2 b) ∩ b) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ b)
3		ancom 68	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∩ b) = (b ∩ (b ∪ (a⊥ ∩ b⊥ )))
4		a5c 113	. . 3 (b ∩ (b ∪ (a⊥ ∩ b⊥ ))) = b
5	3, 4	ax-r2 35	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∩ b) = b
6	2, 5	ax-r2 35	1 ((a →2 b) ∩ b) = b

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 14

This theorem is referenced by:  u2lemnonb 658  u21lembi 709  bi3 821  bi4 822

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i2 44

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