Theorem u3lem14mp 794

Description: Used to prove →₁ modus ponens rule in →₃ system. (Contributed by NM, 19-Jan-1998.)

Assertion

Ref	Expression
u3lem14mp	((a →3 b⊥ )⊥ →3 (a →3 (a →3 b))) = 1

Proof of Theorem u3lem14mp

Step	Hyp	Ref	Expression
1		lear 161	. . . 4 (((a ∪ b⊥ ⊥ ) ∩ (a ∪ b⊥ )) ∩ (a⊥ ∪ (a ∩ b⊥ ⊥ ))) ≤ (a⊥ ∪ (a ∩ b⊥ ⊥ ))
2		lear 161	. . . . . 6 (a ∩ b⊥ ⊥ ) ≤ b⊥ ⊥
3		ax-a1 30	. . . . . . 7 b = b⊥ ⊥
4	3	ax-r1 35	. . . . . 6 b⊥ ⊥ = b
5	2, 4	lbtr 139	. . . . 5 (a ∩ b⊥ ⊥ ) ≤ b
6	5	lelor 166	. . . 4 (a⊥ ∪ (a ∩ b⊥ ⊥ )) ≤ (a⊥ ∪ b)
7	1, 6	letr 137	. . 3 (((a ∪ b⊥ ⊥ ) ∩ (a ∪ b⊥ )) ∩ (a⊥ ∪ (a ∩ b⊥ ⊥ ))) ≤ (a⊥ ∪ b)
8		ud3lem0c 279	. . 3 (a →3 b⊥ )⊥ = (((a ∪ b⊥ ⊥ ) ∩ (a ∪ b⊥ )) ∩ (a⊥ ∪ (a ∩ b⊥ ⊥ )))
9		u3lem5 763	. . 3 (a →3 (a →3 b)) = (a⊥ ∪ b)
10	7, 8, 9	le3tr1 140	. 2 (a →3 b⊥ )⊥ ≤ (a →3 (a →3 b))
11	10	u3lemle1 712	1 ((a →3 b⊥ )⊥ →3 (a →3 (a →3 b))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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: (None)