Theorem pm5.21ni 378

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 205

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 206

This theorem is referenced by:  pm5.21nii  379  norbi  883  pm5.54  1014  niabn  1017  ab0w  4304  csbprc  4337  ralf0  4441  ordsssuc2  6339  ndmovord  7440  ordsucelsuc  7644  brdomg  8703  suppeqfsuppbi  9072  funsnfsupp  9082  r1pw  9534  r1pwALT  9535  elixx3g  13021  elfz2  13175  bifald  36172  areaquad  40963