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Theorem oa4to6dual 964
Description: Lemma for orthoarguesian law (4-variable to 6-variable proof).
Hypotheses
Ref Expression
oa4to6lem.1 a' =< b
oa4to6lem.2 c' =< d
oa4to6lem.3 e' =< f
oa4to6lem.4 g = (((a ^ b) v (c ^ d)) v (e ^ f))
oa4to6lem.oa4 ((a ->1 g) ^ (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))))))) =< g
Assertion
Ref Expression
oa4to6dual (b ^ (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))))) =< g
Proof of Theorem oa4to6dual
StepHypRef Expression
1 oa4to6lem.1 . . 3 a' =< b
2 oa4to6lem.2 . . 3 c' =< d
3 oa4to6lem.3 . . 3 e' =< f
4 oa4to6lem.4 . . 3 g = (((a ^ b) v (c ^ d)) v (e ^ f))
51, 2, 3, 4oa4to6lem4 963 . 2 (b ^ (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))))) =< ((a ->1 g) ^ (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))))))
6 oa4to6lem.oa4 . 2 ((a ->1 g) ^ (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))))))) =< g
75, 6letr 137 1 (b ^ (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))))) =< g
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  oa4to6  965  oa3-6to3  987  oa3-2to4  988  oa3-u1  991  oa3-u2  992
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