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Theorem rp-fakeimass 36773
Description: A special case where implication appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
Assertion
Ref Expression
rp-fakeimass ((𝜑𝜒) ↔ (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))

Proof of Theorem rp-fakeimass
StepHypRef Expression
1 ax-1 6 . . . . . . . 8 (𝜓 → (𝜑𝜓))
21con3i 148 . . . . . . 7 (¬ (𝜑𝜓) → ¬ 𝜓)
32pm2.21d 116 . . . . . 6 (¬ (𝜑𝜓) → (𝜓𝜒))
43a1d 25 . . . . 5 (¬ (𝜑𝜓) → (𝜑 → (𝜓𝜒)))
5 ax-1 6 . . . . . 6 (𝜒 → (𝜓𝜒))
65a1d 25 . . . . 5 (𝜒 → (𝜑 → (𝜓𝜒)))
74, 6ja 171 . . . 4 (((𝜑𝜓) → 𝜒) → (𝜑 → (𝜓𝜒)))
8 ax-2 7 . . . . 5 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
98com3r 84 . . . 4 (𝜑 → ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → 𝜒)))
107, 9impbid2 214 . . 3 (𝜑 → (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
11 ax-1 6 . . . 4 (𝜒 → ((𝜑𝜓) → 𝜒))
1211, 62thd 253 . . 3 (𝜒 → (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
1310, 12jaoi 392 . 2 ((𝜑𝜒) → (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
14 jarl 173 . . . . 5 (((𝜑𝜓) → 𝜒) → (¬ 𝜑𝜒))
1514orrd 391 . . . 4 (((𝜑𝜓) → 𝜒) → (𝜑𝜒))
1615a1d 25 . . 3 (((𝜑𝜓) → 𝜒) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
17 simplim 161 . . . . 5 (¬ (𝜑 → (𝜓𝜒)) → 𝜑)
1817orcd 405 . . . 4 (¬ (𝜑 → (𝜓𝜒)) → (𝜑𝜒))
1918a1i 11 . . 3 (¬ ((𝜑𝜓) → 𝜒) → (¬ (𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
2016, 19bija 368 . 2 ((((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))) → (𝜑𝜒))
2113, 20impbii 197 1 ((𝜑𝜒) ↔ (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381
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 195  df-or 383
This theorem is referenced by: (None)
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