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Theorem oa3to4lem5 949
Description: Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable proof).
Hypothesis
Ref Expression
oa3to4lem5.1 ((a v b) ^ (c v d)) =< (a v (b ^ (d v ((a v c) ^ (b v d)))))
Assertion
Ref Expression
oa3to4lem5 ((b v a) ^ (d v c)) =< (a v (b ^ (d v ((b v d) ^ (a v c)))))
Proof of Theorem oa3to4lem5
StepHypRef Expression
1 oa3to4lem5.1 . 2 ((a v b) ^ (c v d)) =< (a v (b ^ (d v ((a v c) ^ (b v d)))))
2 ax-a2 31 . . 3 (b v a) = (a v b)
3 ax-a2 31 . . 3 (d v c) = (c v d)
42, 32an 79 . 2 ((b v a) ^ (d v c)) = ((a v b) ^ (c v d))
5 ancom 74 . . . . 5 ((b v d) ^ (a v c)) = ((a v c) ^ (b v d))
65lor 70 . . . 4 (d v ((b v d) ^ (a v c))) = (d v ((a v c) ^ (b v d)))
76lan 77 . . 3 (b ^ (d v ((b v d) ^ (a v c)))) = (b ^ (d v ((a v c) ^ (b v d))))
87lor 70 . 2 (a v (b ^ (d v ((b v d) ^ (a v c))))) = (a v (b ^ (d v ((a v c) ^ (b v d)))))
91, 4, 8le3tr1 140 1 ((b v a) ^ (d v c)) =< (a v (b ^ (d v ((b v d) ^ (a v c)))))
Colors of variables: term
Syntax hints:   =< wle 2   v wo 6   ^ wa 7
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-le1 130  df-le2 131
This theorem is referenced by:  oa3to4  951
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