Theorem pm4.11 400

Description: Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117.

Assertion

Ref	Expression
pm4.11	⊢ ((φ ↔ ψ) ↔ (¬ φ ↔ ¬ ψ))

Proof of Theorem pm4.11

Step	Hyp	Ref	Expression
1		pm4.1 143	. . . 4 ⊢ ((φ → ψ) ↔ (¬ ψ → ¬ φ))
2		pm4.1 143	. . . 4 ⊢ ((ψ → φ) ↔ (¬ φ → ¬ ψ))
3	1, 2	anbi12i 369	. . 3 ⊢ (((φ → ψ) ∧ (ψ → φ)) ↔ ((¬ ψ → ¬ φ) ∧ (¬ φ → ¬ ψ)))
4		bi 396	. . 3 ⊢ ((φ ↔ ψ) ↔ ((φ → ψ) ∧ (ψ → φ)))
5		bi 396	. . 3 ⊢ ((¬ ψ ↔ ¬ φ) ↔ ((¬ ψ → ¬ φ) ∧ (¬ φ → ¬ ψ)))
6	3, 4, 5	3bitr4 158	. 2 ⊢ ((φ ↔ ψ) ↔ (¬ ψ ↔ ¬ φ))
7		bicom 398	. 2 ⊢ ((¬ ψ ↔ ¬ φ) ↔ (¬ φ ↔ ¬ ψ))
8	6, 7	bitr 151	1 ⊢ ((φ ↔ ψ) ↔ (¬ φ ↔ ¬ ψ))

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

This theorem is referenced by:  bicon4i 401  bicon4d 402  negbid 463  pm5.32 488  cbvexd 978

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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