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Theorem bilukdc 1284
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 φ DECID ψ) DECID χ) → ((φψ) ↔ ((χψ) ↔ (φχ))))

Proof of Theorem bilukdc
StepHypRef Expression
1 bicom 128 . . . . . 6 ((φψ) ↔ (ψφ))
21bibi1i 217 . . . . 5 (((φψ) ↔ χ) ↔ ((ψφ) ↔ χ))
3 biassdc 1283 . . . . . 6 (DECID ψ → (DECID φ → (DECID χ → (((ψφ) ↔ χ) ↔ (ψ ↔ (φχ))))))
43imp31 243 . . . . 5 (((DECID ψ DECID φ) DECID χ) → (((ψφ) ↔ χ) ↔ (ψ ↔ (φχ))))
52, 4syl5bb 181 . . . 4 (((DECID ψ DECID φ) DECID χ) → (((φψ) ↔ χ) ↔ (ψ ↔ (φχ))))
65ancom1s 503 . . 3 (((DECID φ DECID ψ) DECID χ) → (((φψ) ↔ χ) ↔ (ψ ↔ (φχ))))
7 dcbi 843 . . . . . 6 (DECID φ → (DECID ψDECID (φψ)))
87imp 115 . . . . 5 ((DECID φ DECID ψ) → DECID (φψ))
98adantr 261 . . . 4 (((DECID φ DECID ψ) DECID χ) → DECID (φψ))
10 simpr 103 . . . 4 (((DECID φ DECID ψ) DECID χ) → DECID χ)
11 dcbi 843 . . . . . 6 (DECID φ → (DECID χDECID (φχ)))
12 dcbi 843 . . . . . 6 (DECID ψ → (DECID (φχ) → DECID (ψ ↔ (φχ))))
1311, 12syl9 66 . . . . 5 (DECID φ → (DECID ψ → (DECID χDECID (ψ ↔ (φχ)))))
1413imp31 243 . . . 4 (((DECID φ DECID ψ) DECID χ) → DECID (ψ ↔ (φχ)))
15 biassdc 1283 . . . 4 (DECID (φψ) → (DECID χ → (DECID (ψ ↔ (φχ)) → ((((φψ) ↔ χ) ↔ (ψ ↔ (φχ))) ↔ ((φψ) ↔ (χ ↔ (ψ ↔ (φχ))))))))
169, 10, 14, 15syl3c 57 . . 3 (((DECID φ DECID ψ) DECID χ) → ((((φψ) ↔ χ) ↔ (ψ ↔ (φχ))) ↔ ((φψ) ↔ (χ ↔ (ψ ↔ (φχ))))))
176, 16mpbid 135 . 2 (((DECID φ DECID ψ) DECID χ) → ((φψ) ↔ (χ ↔ (ψ ↔ (φχ)))))
18 simplr 482 . . 3 (((DECID φ DECID ψ) DECID χ) → DECID ψ)
1911imp 115 . . . 4 ((DECID φ DECID χ) → DECID (φχ))
2019adantlr 446 . . 3 (((DECID φ DECID ψ) DECID χ) → DECID (φχ))
21 biassdc 1283 . . 3 (DECID χ → (DECID ψ → (DECID (φχ) → (((χψ) ↔ (φχ)) ↔ (χ ↔ (ψ ↔ (φχ)))))))
2210, 18, 20, 21syl3c 57 . 2 (((DECID φ DECID ψ) DECID χ) → (((χψ) ↔ (φχ)) ↔ (χ ↔ (ψ ↔ (φχ)))))
2317, 22bitr4d 180 1 (((DECID φ DECID ψ) DECID χ) → ((φψ) ↔ ((χψ) ↔ (φχ))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  DECID wdc 741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by: (None)
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