Theorem wlebi 384

Description: L.e. to biconditional.

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 361	. . . 4 ((b ∪ a) ≡ a) = 1
3	2	wr1 189	. . 3 (a ≡ (b ∪ a)) = 1
4		ax-a2 30	. . . 4 (b ∪ a) = (a ∪ b)
5	4	bi1 110	. . 3 ((b ∪ a) ≡ (a ∪ b)) = 1
6	3, 5	wr2 353	. 2 (a ≡ (a ∪ b)) = 1
7		wlebi.1	. . 3 (a ≤2 b) = 1
8	7	wdf-le2 361	. 2 ((a ∪ b) ≡ b) = 1
9	6, 8	wr2 353	1 (a ≡ b) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6  1wt 9   ≤₂ wle2 11

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-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le 121  df-le1 122  df-le2 123

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