Theorem mtt 534

Description: Modus-tollens-like theorem.

Assertion

Ref	Expression
mtt	⊢ (¬ φ → (¬ ψ ↔ (ψ → φ)))

Proof of Theorem mtt

Step	Hyp	Ref	Expression
1		pm2.21 71	. . 3 ⊢ (¬ ψ → (ψ → φ))
2	1	a1i 7	. 2 ⊢ (¬ φ → (¬ ψ → (ψ → φ)))
3		con3 86	. . 3 ⊢ ((ψ → φ) → (¬ φ → ¬ ψ))
4	3	com12 13	. 2 ⊢ (¬ φ → ((ψ → φ) → ¬ ψ))
5	2, 4	impbid 397	1 ⊢ (¬ φ → (¬ ψ ↔ (ψ → φ)))

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

This theorem is referenced by:  axpowndlem3 3745  axpownd 3747  large 5700

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

metamath.org