Theorem bicon2 403

Description: Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117.

Assertion

Ref	Expression
bicon2	⊢ ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ))

Proof of Theorem bicon2

Step	Hyp	Ref	Expression
1		bi2.03 144	. . 3 ⊢ ((φ → ¬ ψ) ↔ (ψ → ¬ φ))
2		bi2.15 145	. . 3 ⊢ ((¬ ψ → φ) ↔ (¬ φ → ψ))
3	1, 2	anbi12i 369	. 2 ⊢ (((φ → ¬ ψ) ∧ (¬ ψ → φ)) ↔ ((ψ → ¬ φ) ∧ (¬ φ → ψ)))
4		bi 396	. 2 ⊢ ((φ ↔ ¬ ψ) ↔ ((φ → ¬ ψ) ∧ (¬ ψ → φ)))
5		bi 396	. 2 ⊢ ((ψ ↔ ¬ φ) ↔ ((ψ → ¬ φ) ∧ (¬ φ → ψ)))
6	3, 4, 5	3bitr4 158	1 ⊢ ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ))

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

This theorem is referenced by:  bicon2d 404

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-an 198

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