Theorem wlebi 402

Description: L.e. to biconditional. (Contributed by NM, 13-Oct-1997.)

Hypotheses

Ref	Expression
wlebi.1	(a ≤2 b) = 1
wlebi.2	(b ≤2 a) = 1

Assertion

Ref	Expression
wlebi	(a ≡ b) = 1

Proof of Theorem wlebi

Step	Hyp	Ref	Expression
1		wlebi.2	. . . . 5 (b ≤2 a) = 1
2	1	wdf-le2 379	. . . 4 ((b ∪ a) ≡ a) = 1
3	2	wr1 197	. . 3 (a ≡ (b ∪ a)) = 1
4		ax-a2 31	. . . 4 (b ∪ a) = (a ∪ b)
5	4	bi1 118	. . 3 ((b ∪ a) ≡ (a ∪ b)) = 1
6	3, 5	wr2 371	. 2 (a ≡ (a ∪ b)) = 1
7		wlebi.1	. . 3 (a ≤2 b) = 1
8	7	wdf-le2 379	. 2 ((a ∪ b) ≡ b) = 1
9	6, 8	wr2 371	1 (a ≡ b) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6  1wt 8   ≤₂ wle2 10

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-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131

This theorem is referenced by: (None)