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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  pm5.21nii  381  norbi  900  pm5.54  1035  niabn  1038  sbccomlem  3816  csbprc  4366  ordsssuc2  6456  ndmovord  7612  ordsucelsuc  7834  brdomg  8989  suppeqfsuppbi  9375  funsnfsupp  9388  r1pw  9863  r1pwALT  9864  elixx3g  13493  elfz2  13650  bifald  39115  quadfac  43349  areaquad  44312