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Theorem oa3-2lemb 979
Description: Lemma for 3-OA(2). Equivalence with substitution into 4-OA.
Assertion
Ref Expression
oa3-2lemb ((a ->1 c) ^ (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c)))))))) = ((a ->1 c) ^ (a v (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))))
Proof of Theorem oa3-2lemb
StepHypRef Expression
1 ax-a3 32 . . . . 5 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))) = ((a ^ b) v (((a ->1 c) ^ (b ->1 c)) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))))
2 i1id 275 . . . . . . . . . . . . 13 (c ->1 c) = 1
32lan 77 . . . . . . . . . . . 12 ((a ->1 c) ^ (c ->1 c)) = ((a ->1 c) ^ 1)
4 an1 106 . . . . . . . . . . . 12 ((a ->1 c) ^ 1) = (a ->1 c)
53, 4ax-r2 36 . . . . . . . . . . 11 ((a ->1 c) ^ (c ->1 c)) = (a ->1 c)
65lor 70 . . . . . . . . . 10 ((a ^ c) v ((a ->1 c) ^ (c ->1 c))) = ((a ^ c) v (a ->1 c))
7 or12 80 . . . . . . . . . . . 12 ((a ^ c) v (a' v (a ^ c))) = (a' v ((a ^ c) v (a ^ c)))
8 oridm 110 . . . . . . . . . . . . 13 ((a ^ c) v (a ^ c)) = (a ^ c)
98lor 70 . . . . . . . . . . . 12 (a' v ((a ^ c) v (a ^ c))) = (a' v (a ^ c))
107, 9ax-r2 36 . . . . . . . . . . 11 ((a ^ c) v (a' v (a ^ c))) = (a' v (a ^ c))
11 df-i1 44 . . . . . . . . . . . 12 (a ->1 c) = (a' v (a ^ c))
1211lor 70 . . . . . . . . . . 11 ((a ^ c) v (a ->1 c)) = ((a ^ c) v (a' v (a ^ c)))
1310, 12, 113tr1 63 . . . . . . . . . 10 ((a ^ c) v (a ->1 c)) = (a ->1 c)
146, 13ax-r2 36 . . . . . . . . 9 ((a ^ c) v ((a ->1 c) ^ (c ->1 c))) = (a ->1 c)
152lan 77 . . . . . . . . . . . 12 ((b ->1 c) ^ (c ->1 c)) = ((b ->1 c) ^ 1)
16 an1 106 . . . . . . . . . . . 12 ((b ->1 c) ^ 1) = (b ->1 c)
1715, 16ax-r2 36 . . . . . . . . . . 11 ((b ->1 c) ^ (c ->1 c)) = (b ->1 c)
1817lor 70 . . . . . . . . . 10 ((b ^ c) v ((b ->1 c) ^ (c ->1 c))) = ((b ^ c) v (b ->1 c))
19 or12 80 . . . . . . . . . . . 12 ((b ^ c) v (b' v (b ^ c))) = (b' v ((b ^ c) v (b ^ c)))
20 oridm 110 . . . . . . . . . . . . 13 ((b ^ c) v (b ^ c)) = (b ^ c)
2120lor 70 . . . . . . . . . . . 12 (b' v ((b ^ c) v (b ^ c))) = (b' v (b ^ c))
2219, 21ax-r2 36 . . . . . . . . . . 11 ((b ^ c) v (b' v (b ^ c))) = (b' v (b ^ c))
23 df-i1 44 . . . . . . . . . . . 12 (b ->1 c) = (b' v (b ^ c))
2423lor 70 . . . . . . . . . . 11 ((b ^ c) v (b ->1 c)) = ((b ^ c) v (b' v (b ^ c)))
2522, 24, 233tr1 63 . . . . . . . . . 10 ((b ^ c) v (b ->1 c)) = (b ->1 c)
2618, 25ax-r2 36 . . . . . . . . 9 ((b ^ c) v ((b ->1 c) ^ (c ->1 c))) = (b ->1 c)
2714, 262an 79 . . . . . . . 8 (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c)))) = ((a ->1 c) ^ (b ->1 c))
2827lor 70 . . . . . . 7 (((a ->1 c) ^ (b ->1 c)) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))) = (((a ->1 c) ^ (b ->1 c)) v ((a ->1 c) ^ (b ->1 c)))
29 oridm 110 . . . . . . 7 (((a ->1 c) ^ (b ->1 c)) v ((a ->1 c) ^ (b ->1 c))) = ((a ->1 c) ^ (b ->1 c))
3028, 29ax-r2 36 . . . . . 6 (((a ->1 c) ^ (b ->1 c)) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))) = ((a ->1 c) ^ (b ->1 c))
3130lor 70 . . . . 5 ((a ^ b) v (((a ->1 c) ^ (b ->1 c)) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c)))))) = ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
321, 31ax-r2 36 . . . 4 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))) = ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
3332lan 77 . . 3 (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c)))))) = (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))
3433lor 70 . 2 (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))))) = (a v (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))))
3534lan 77 1 ((a ->1 c) ^ (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c)))))))) = ((a ->1 c) ^ (a v (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44
This theorem is referenced by:  oa3-2to4  988
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