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Theorem con3th 924
 Description: Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version of con3 126 demonstrates the use of the weak deduction theorem dedt 923 to derive it from con3i 127. (Contributed by NM, 27-Jun-2002.) (Proof modification is discouraged.)
Assertion
Ref Expression
con3th ((φψ) → (¬ ψ → ¬ φ))

Proof of Theorem con3th
StepHypRef Expression
1 id 19 . . . 4 ((ψ ↔ ((ψ (φψ)) (φ ¬ (φψ)))) → (ψ ↔ ((ψ (φψ)) (φ ¬ (φψ)))))
21notbid 285 . . 3 ((ψ ↔ ((ψ (φψ)) (φ ¬ (φψ)))) → (¬ ψ ↔ ¬ ((ψ (φψ)) (φ ¬ (φψ)))))
32imbi1d 308 . 2 ((ψ ↔ ((ψ (φψ)) (φ ¬ (φψ)))) → ((¬ ψ → ¬ φ) ↔ (¬ ((ψ (φψ)) (φ ¬ (φψ))) → ¬ φ)))
41imbi2d 307 . . . 4 ((ψ ↔ ((ψ (φψ)) (φ ¬ (φψ)))) → ((φψ) ↔ (φ → ((ψ (φψ)) (φ ¬ (φψ))))))
5 id 19 . . . . 5 ((φ ↔ ((ψ (φψ)) (φ ¬ (φψ)))) → (φ ↔ ((ψ (φψ)) (φ ¬ (φψ)))))
65imbi2d 307 . . . 4 ((φ ↔ ((ψ (φψ)) (φ ¬ (φψ)))) → ((φφ) ↔ (φ → ((ψ (φψ)) (φ ¬ (φψ))))))
7 id 19 . . . 4 (φφ)
84, 6, 7elimh 922 . . 3 (φ → ((ψ (φψ)) (φ ¬ (φψ))))
98con3i 127 . 2 (¬ ((ψ (φψ)) (φ ¬ (φψ))) → ¬ φ)
103, 9dedt 923 1 ((φψ) → (¬ ψ → ¬ φ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358 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 177  df-or 359  df-an 360 This theorem is referenced by: (None)
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