Theorem pm5.21ni 377

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 376	1 ⊢ (¬ 𝜓 → (𝜑 ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206

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 207

This theorem is referenced by:  pm5.21nii  378  norbi  885  pm5.54  1018  niabn  1021  sbccomlem  3891  ab0w  4401  csbprc  4432  ralf0  4537  ordsssuc2  6486  ndmovord  7640  ordsucelsuc  7858  brdomg  9016  brdomgOLD  9017  suppeqfsuppbi  9448  funsnfsupp  9461  r1pw  9914  r1pwALT  9915  elixx3g  13420  elfz2  13574  bifald  38047  areaquad  43177