Theorem pm4.1 164

Description: Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116.

Assertion

Ref	Expression
pm4.1	⊢ ((φ → ψ) ↔ (¬ ψ → ¬ φ))

Proof of Theorem pm4.1

Step	Hyp	Ref	Expression
1		con3 94	. 2 ⊢ ((φ → ψ) → (¬ ψ → ¬ φ))
2		ax-3 6	. 2 ⊢ ((¬ ψ → ¬ φ) → (φ → ψ))
3	1, 2	impbi 157	1 ⊢ ((φ → ψ) ↔ (¬ ψ → ¬ φ))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146

This theorem is referenced by:  pm4.79 355  pm4.11 521  imbi1d 612  dfom2 3128  indstr 6401  ntreq0 7658

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147

Copyright terms: Public domain