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Theorem bilukdc 1328
Description: Lukasiewicz's shortest axiom for equivalential calculus (but modified to require decidable propositions). Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
bilukdc (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → ((𝜑𝜓) ↔ ((𝜒𝜓) ↔ (𝜑𝜒))))

Proof of Theorem bilukdc
StepHypRef Expression
1 bicom 138 . . . . . 6 ((𝜑𝜓) ↔ (𝜓𝜑))
21bibi1i 226 . . . . 5 (((𝜑𝜓) ↔ 𝜒) ↔ ((𝜓𝜑) ↔ 𝜒))
3 biassdc 1327 . . . . . 6 (DECID 𝜓 → (DECID 𝜑 → (DECID 𝜒 → (((𝜓𝜑) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑𝜒))))))
43imp31 252 . . . . 5 (((DECID 𝜓DECID 𝜑) ∧ DECID 𝜒) → (((𝜓𝜑) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑𝜒))))
52, 4syl5bb 190 . . . 4 (((DECID 𝜓DECID 𝜑) ∧ DECID 𝜒) → (((𝜑𝜓) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑𝜒))))
65ancom1s 534 . . 3 (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → (((𝜑𝜓) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑𝜒))))
7 dcbi 878 . . . . . 6 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
87imp 122 . . . . 5 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
98adantr 270 . . . 4 (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → DECID (𝜑𝜓))
10 simpr 108 . . . 4 (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → DECID 𝜒)
11 dcbi 878 . . . . . 6 (DECID 𝜑 → (DECID 𝜒DECID (𝜑𝜒)))
12 dcbi 878 . . . . . 6 (DECID 𝜓 → (DECID (𝜑𝜒) → DECID (𝜓 ↔ (𝜑𝜒))))
1311, 12syl9 71 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒DECID (𝜓 ↔ (𝜑𝜒)))))
1413imp31 252 . . . 4 (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → DECID (𝜓 ↔ (𝜑𝜒)))
15 biassdc 1327 . . . 4 (DECID (𝜑𝜓) → (DECID 𝜒 → (DECID (𝜓 ↔ (𝜑𝜒)) → ((((𝜑𝜓) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑𝜒))) ↔ ((𝜑𝜓) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑𝜒))))))))
169, 10, 14, 15syl3c 62 . . 3 (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → ((((𝜑𝜓) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑𝜒))) ↔ ((𝜑𝜓) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑𝜒))))))
176, 16mpbid 145 . 2 (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → ((𝜑𝜓) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑𝜒)))))
18 simplr 497 . . 3 (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → DECID 𝜓)
1911imp 122 . . . 4 ((DECID 𝜑DECID 𝜒) → DECID (𝜑𝜒))
2019adantlr 461 . . 3 (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → DECID (𝜑𝜒))
21 biassdc 1327 . . 3 (DECID 𝜒 → (DECID 𝜓 → (DECID (𝜑𝜒) → (((𝜒𝜓) ↔ (𝜑𝜒)) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑𝜒)))))))
2210, 18, 20, 21syl3c 62 . 2 (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → (((𝜒𝜓) ↔ (𝜑𝜒)) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑𝜒)))))
2317, 22bitr4d 189 1 (((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → ((𝜑𝜓) ↔ ((𝜒𝜓) ↔ (𝜑𝜒))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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