Theorem pm5.21ni 341

Description: Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Hypotheses

Ref	Expression
pm5.21ni.1	⊢ (φ → ψ)
pm5.21ni.2	⊢ (χ → ψ)

Assertion

Ref	Expression
pm5.21ni	⊢ (¬ ψ → (φ ↔ χ))

Proof of Theorem pm5.21ni

Step	Hyp	Ref	Expression
1		pm5.21ni.1	. . 3 ⊢ (φ → ψ)
2	1	con3i 127	. 2 ⊢ (¬ ψ → ¬ φ)
3		pm5.21ni.2	. . 3 ⊢ (χ → ψ)
4	3	con3i 127	. 2 ⊢ (¬ ψ → ¬ χ)
5	2, 4	2falsed 340	1 ⊢ (¬ ψ → (φ ↔ χ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  pm5.21nii  342  pm5.54  870  niabn  917  ndmovord  5621