Theorem binr3 519

Description: Pavicic binary logic ax-r3 439 analog. (Contributed by NM, 7-Nov-1997.)

Hypotheses

Ref	Expression
binr3.1	(a →3 c) = 1
binr3.2	(b →3 c) = 1

Assertion

Ref	Expression
binr3	((a ∪ b) →3 c) = 1

Proof of Theorem binr3

Step	Hyp	Ref	Expression
1		binr3.1	. . . . 5 (a →3 c) = 1
2	1	i3le 515	. . . 4 a ≤ c
3		binr3.2	. . . . 5 (b →3 c) = 1
4	3	i3le 515	. . . 4 b ≤ c
5	2, 4	le2or 168	. . 3 (a ∪ b) ≤ (c ∪ c)
6		oridm 110	. . 3 (c ∪ c) = c
7	5, 6	lbtr 139	. 2 (a ∪ b) ≤ c
8	7	lei3 246	1 ((a ∪ b) →3 c) = 1

Syntax hints:   = wb 1   ∪ wo 6  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:  i3ror  532