Theorem gomaex3h10 891

Description: Hypothesis for Godowski 6-var -> Mayet Example 3.

Hypotheses

Ref	Expression
gomaex3h10.10	q = ((e v f) ->1 (b v c)_|_)_|_
gomaex3h10.21	x = q
gomaex3h10.22	y = (e v f)_|_

Assertion

Ref	Expression
gomaex3h10	x =< y_|_

Proof of Theorem gomaex3h10

Step	Hyp	Ref	Expression
1		lea 152	. . 3 ((e v f) ^ ((e v f) ^ (b v c)_|_)_|_) =< (e v f)
2		gomaex3h10.10	. . . 4 q = ((e v f) ->1 (b v c)_|_)_|_
3		df-i1 43	. . . . . 6 ((e v f) ->1 (b v c)_|_) = ((e v f)_|_ v ((e v f) ^ (b v c)_|_))
4	3	ax-r4 36	. . . . 5 ((e v f) ->1 (b v c)_|_)_|_ = ((e v f)_|_ v ((e v f) ^ (b v c)_|_))_|_
5		anor1 80	. . . . . 6 ((e v f) ^ ((e v f) ^ (b v c)_|_)_|_) = ((e v f)_|_ v ((e v f) ^ (b v c)_|_))_|_
6	5	ax-r1 34	. . . . 5 ((e v f)_|_ v ((e v f) ^ (b v c)_|_))_|_ = ((e v f) ^ ((e v f) ^ (b v c)_|_)_|_)
7	4, 6	ax-r2 35	. . . 4 ((e v f) ->1 (b v c)_|_)_|_ = ((e v f) ^ ((e v f) ^ (b v c)_|_)_|_)
8	2, 7	ax-r2 35	. . 3 q = ((e v f) ^ ((e v f) ^ (b v c)_|_)_|_)
9		ax-a1 29	. . . 4 (e v f) = (e v f)_|__|_
10	9	ax-r1 34	. . 3 (e v f)_|__|_ = (e v f)
11	1, 8, 10	le3tr1 132	. 2 q =< (e v f)_|__|_
12		gomaex3h10.21	. 2 x = q
13		gomaex3h10.22	. . 3 y = (e v f)_|_
14	13	ax-r4 36	. 2 y_|_ = (e v f)_|__|_
15	11, 12, 14	le3tr1 132	1 x =< y_|_

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13

This theorem is referenced by:  gomaex3lem5 898

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i1 43  df-le1 122  df-le2 123

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