Theorem u3lem14aa2 793

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

Assertion

Ref	Expression
u3lem14aa2	(a →3 (a →3 (b →3 (b →3 a⊥ )⊥ ))) = 1

Proof of Theorem u3lem14aa2

Step	Hyp	Ref	Expression
1		u3lem13a 789	. . . . 5 (b →3 (b →3 a⊥ )⊥ ) = (b →1 a)
2		u3lem13b 790	. . . . . 6 ((b →3 a⊥ ) →3 b⊥ ) = (b →1 a)
3	2	ax-r1 35	. . . . 5 (b →1 a) = ((b →3 a⊥ ) →3 b⊥ )
4	1, 3	ax-r2 36	. . . 4 (b →3 (b →3 a⊥ )⊥ ) = ((b →3 a⊥ ) →3 b⊥ )
5	4	ud3lem0a 260	. . 3 (a →3 (b →3 (b →3 a⊥ )⊥ )) = (a →3 ((b →3 a⊥ ) →3 b⊥ ))
6	5	ud3lem0a 260	. 2 (a →3 (a →3 (b →3 (b →3 a⊥ )⊥ ))) = (a →3 (a →3 ((b →3 a⊥ ) →3 b⊥ )))
7		u3lem14aa 792	. 2 (a →3 (a →3 ((b →3 a⊥ ) →3 b⊥ ))) = 1
8	6, 7	ax-r2 36	1 (a →3 (a →3 (b →3 (b →3 a⊥ )⊥ ))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4  1wt 8   →₁ wi1 12   →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-i1 44  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)