Theorem mt2bi 535

Description: A false consequent falsifies an antecedent.

Hypothesis

Ref	Expression
mt2bi.1	⊢ φ

Assertion

Ref	Expression
mt2bi	⊢ (¬ ψ ↔ (ψ → ¬ φ))

Proof of Theorem mt2bi

Step	Hyp	Ref	Expression
1		pm2.21 71	. 2 ⊢ (¬ ψ → (ψ → ¬ φ))
2		mt2bi.1	. . 3 ⊢ φ
3		con2 82	. . 3 ⊢ ((ψ → ¬ φ) → (φ → ¬ ψ))
4	2, 3	mpi 44	. 2 ⊢ ((ψ → ¬ φ) → ¬ ψ)
5	1, 4	impbi 139	1 ⊢ (¬ ψ ↔ (ψ → ¬ φ))

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

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

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