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Theorem rp-fakeimass 40137
 Description: A special case where implication appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)
Assertion
Ref Expression
rp-fakeimass ((𝜑𝜒) ↔ (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))

Proof of Theorem rp-fakeimass
StepHypRef Expression
1 conax1 173 . . . . . . 7 (¬ (𝜑𝜓) → ¬ 𝜓)
21pm2.21d 121 . . . . . 6 (¬ (𝜑𝜓) → (𝜓𝜒))
32a1d 25 . . . . 5 (¬ (𝜑𝜓) → (𝜑 → (𝜓𝜒)))
4 ax-1 6 . . . . . 6 (𝜒 → (𝜓𝜒))
54a1d 25 . . . . 5 (𝜒 → (𝜑 → (𝜓𝜒)))
63, 5ja 189 . . . 4 (((𝜑𝜓) → 𝜒) → (𝜑 → (𝜓𝜒)))
7 ax-2 7 . . . . 5 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
87com3r 87 . . . 4 (𝜑 → ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → 𝜒)))
96, 8impbid2 229 . . 3 (𝜑 → (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
10 ax-1 6 . . . 4 (𝜒 → ((𝜑𝜓) → 𝜒))
1110, 52thd 268 . . 3 (𝜒 → (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
129, 11jaoi 854 . 2 ((𝜑𝜒) → (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
13 jarl 125 . . . . 5 (((𝜑𝜓) → 𝜒) → (¬ 𝜑𝜒))
1413orrd 860 . . . 4 (((𝜑𝜓) → 𝜒) → (𝜑𝜒))
1514a1d 25 . . 3 (((𝜑𝜓) → 𝜒) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
16 simplim 170 . . . . 5 (¬ (𝜑 → (𝜓𝜒)) → 𝜑)
1716orcd 870 . . . 4 (¬ (𝜑 → (𝜓𝜒)) → (𝜑𝜒))
1817a1i 11 . . 3 (¬ ((𝜑𝜓) → 𝜒) → (¬ (𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
1915, 18bija 385 . 2 ((((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))) → (𝜑𝜒))
2012, 19impbii 212 1 ((𝜑𝜒) ↔ (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 210  df-or 845 This theorem is referenced by: (None)
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