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Theorem mdasr 15
Description: is associative. Reverse of ax-mdas 9.
Assertion
Ref Expression
mdasr (~ (𝜑 ⅋ (𝜓𝜒)) ⅋ ((𝜑𝜓) ⅋ 𝜒))

Proof of Theorem mdasr
StepHypRef Expression
1 ax-mdco 8 . . . . 5 (~ (𝜑 ⅋ (𝜓𝜒)) ⅋ ((𝜓𝜒) ⅋ 𝜑))
2 ax-mdas 9 . . . . 5 (~ ((𝜓𝜒) ⅋ 𝜑) ⅋ (𝜓 ⅋ (𝜒𝜑)))
31, 2ax-cut 6 . . . 4 (~ (𝜑 ⅋ (𝜓𝜒)) ⅋ (𝜓 ⅋ (𝜒𝜑)))
4 ax-mdco 8 . . . 4 (~ (𝜓 ⅋ (𝜒𝜑)) ⅋ ((𝜒𝜑) ⅋ 𝜓))
53, 4ax-cut 6 . . 3 (~ (𝜑 ⅋ (𝜓𝜒)) ⅋ ((𝜒𝜑) ⅋ 𝜓))
6 ax-mdas 9 . . 3 (~ ((𝜒𝜑) ⅋ 𝜓) ⅋ (𝜒 ⅋ (𝜑𝜓)))
75, 6ax-cut 6 . 2 (~ (𝜑 ⅋ (𝜓𝜒)) ⅋ (𝜒 ⅋ (𝜑𝜓)))
8 ax-mdco 8 . 2 (~ (𝜒 ⅋ (𝜑𝜓)) ⅋ ((𝜑𝜓) ⅋ 𝜒))
97, 8ax-cut 6 1 (~ (𝜑 ⅋ (𝜓𝜒)) ⅋ ((𝜑𝜓) ⅋ 𝜒))
Colors of variables: wff var nilad
Syntax hints:  wmd 2  ~ wneg 3
This theorem was proved from axioms:  ax-cut 6  ax-mdco 8  ax-mdas 9
This theorem is referenced by:  mdasrd  16
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