Theorem binr3 501

Description: Pavicic binary logic axr3 analog.

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 497	. . . 4 a ≤ c
3		binr3.2	. . . . 5 (b →3 c) = 1
4	3	i3le 497	. . . 4 b ≤ c
5	2, 4	le2or 160	. . 3 (a ∪ b) ≤ (c ∪ c)
6		oridm 102	. . 3 (c ∪ c) = c
7	5, 6	lbtr 131	. 2 (a ∪ b) ≤ c
8	7	lei3 238	1 ((a ∪ b) →3 c) = 1

Syntax hints:   = wb 1   ∪ wo 6  1wt 9   →₃ wi3 15

This theorem is referenced by:  i3ror 514

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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