Theorem gomaex3lem4 897

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

Hypothesis

Ref	Expression
gomaex3lem4.9	p = ((a v b) ->1 (d v e)_|_)_|_

Assertion

Ref	Expression
gomaex3lem4	((a v b) ^ (d v e)_|_) =< p_|_

Proof of Theorem gomaex3lem4

Step	Hyp	Ref	Expression
1		leor 151	. 2 ((a v b) ^ (d v e)_|_) =< ((a v b)_|_ v ((a v b) ^ (d v e)_|_))
2		ax-a1 29	. . 3 ((a v b) ->1 (d v e)_|_) = ((a v b) ->1 (d v e)_|_)_|__|_
3		df-i1 43	. . . 4 ((a v b) ->1 (d v e)_|_) = ((a v b)_|_ v ((a v b) ^ (d v e)_|_))
4	3	ax-r1 34	. . 3 ((a v b)_|_ v ((a v b) ^ (d v e)_|_)) = ((a v b) ->1 (d v e)_|_)
5		gomaex3lem4.9	. . . 4 p = ((a v b) ->1 (d v e)_|_)_|_
6	5	ax-r4 36	. . 3 p_|_ = ((a v b) ->1 (d v e)_|_)_|__|_
7	2, 4, 6	3tr1 60	. 2 ((a v b)_|_ v ((a v b) ^ (d v e)_|_)) = p_|_
8	1, 7	lbtr 131	1 ((a v b) ^ (d v e)_|_) =< p_|_

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

This theorem is referenced by:  gomaex3lem9 902

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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