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  887  pm5.54  1020  niabn  1023  sbccomlem  3808  csbprc  4350  ordsssuc2  6410  ndmovord  7550  ordsucelsuc  7766  brdomg  8898  suppeqfsuppbi  9285  funsnfsupp  9298  r1pw  9760  r1pwALT  9761  elixx3g  13302  elfz2  13459  bifald  38422  areaquad  43662