Theorem pm5.21ni 379

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 154	. 2 ⊢ (¬ 𝜓 → ¬ 𝜑)
3		pm5.21ni.2	. . 3 ⊢ (𝜒 → 𝜓)
4	3	con3i 154	. 2 ⊢ (¬ 𝜓 → ¬ 𝜒)
5	2, 4	2falsed 378	1 ⊢ (¬ 𝜓 → (𝜑 ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208

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 209

This theorem is referenced by:  pm5.21nii  380  norbi  897  pm5.54  1030  niabn  1033  sbccomlem  3820  csbprc  4360  ordsssuc2  6434  ndmovord  7581  ordsucelsuc  7797  brdomg  8933  suppeqfsuppbi  9319  funsnfsupp  9332  r1pw  9797  r1pwALT  9798  elixx3g  13356  elfz2  13513  bifald  38547  areaquad  43754