Theorem wlel 374

Description: Add conjunct to left of l.e.

Hypothesis

Ref	Expression
wle.1	(a ≤2 b) = 1

Assertion

Ref	Expression
wlel	((a ∩ c) ≤2 b) = 1

Proof of Theorem wlel

Step	Hyp	Ref	Expression
1		an32 76	. . . 4 ((a ∩ c) ∩ b) = ((a ∩ b) ∩ c)
2	1	bi1 110	. . 3 (((a ∩ c) ∩ b) ≡ ((a ∩ b) ∩ c)) = 1
3		wle.1	. . . . 5 (a ≤2 b) = 1
4	3	wdf2le2 368	. . . 4 ((a ∩ b) ≡ a) = 1
5	4	wran 351	. . 3 (((a ∩ b) ∩ c) ≡ (a ∩ c)) = 1
6	2, 5	wr2 353	. 2 (((a ∩ c) ∩ b) ≡ (a ∩ c)) = 1
7	6	wdf2le1 367	1 ((a ∩ c) ≤2 b) = 1

Syntax hints:   = wb 1   ∩ wa 7  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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