Theorem con3th 573

Description: Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version of con3 86 demonstrates the use of the weak deduction theorem to derive it from con3i 90.

Assertion

Ref	Expression
con3th	⊢ ((φ → ψ) → (¬ ψ → ¬ φ))

Proof of Theorem con3th

Step	Hyp	Ref	Expression
1		id 9	. . . 4 ⊢ ((ψ ↔ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ)))) → (ψ ↔ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ)))))
2	1	negbid 463	. . 3 ⊢ ((ψ ↔ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ)))) → (¬ ψ ↔ ¬ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ)))))
3	2	imbi1d 465	. 2 ⊢ ((ψ ↔ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ)))) → ((¬ ψ → ¬ φ) ↔ (¬ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ))) → ¬ φ)))
4	1	imbi2d 464	. . . 4 ⊢ ((ψ ↔ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ)))) → ((φ → ψ) ↔ (φ → ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ))))))
5		id 9	. . . . 5 ⊢ ((φ ↔ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ)))) → (φ ↔ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ)))))
6	5	imbi2d 464	. . . 4 ⊢ ((φ ↔ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ)))) → ((φ → φ) ↔ (φ → ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ))))))
7		id 9	. . . 4 ⊢ (φ → φ)
8	4, 6, 7	elimh 571	. . 3 ⊢ (φ → ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ))))
9	8	con3i 90	. 2 ⊢ (¬ ((ψ ∧ (φ → ψ)) ∨ (φ ∧ ¬ (φ → ψ))) → ¬ φ)
10	3, 9	dedt 572	1 ⊢ ((φ → ψ) → (¬ ψ → ¬ φ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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