Theorem pm5.21ni 380

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 155	. 2 ⊢ (¬ 𝜓 → ¬ 𝜑)
3		pm5.21ni.2	. . 3 ⊢ (𝜒 → 𝜓)
4	3	con3i 155	. 2 ⊢ (¬ 𝜓 → ¬ 𝜒)
5	2, 4	2falsed 379	1 ⊢ (¬ 𝜓 → (𝜑 ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209

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 210

This theorem is referenced by:  pm5.21nii  381  norbi  899  pm5.54  1035  niabn  1038  sbccomlem  3823  csbprc  4375  ordsssuc2  6456  ndmovord  7602  ordsucelsuc  7819  brdomg  8956  suppeqfsuppbi  9340  funsnfsupp  9353  r1pw  9818  r1pwALT  9819  elixx3g  13386  elfz2  13543  bifald  38719  quadfac  42953  areaquad  43926