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  886  pm5.54  1019  niabn  1022  sbccomlem  3832  ab0w  4342  csbprc  4372  ralf0  4477  ordsssuc2  6425  ndmovord  7579  ordsucelsuc  7797  brdomg  8930  suppeqfsuppbi  9330  funsnfsupp  9343  r1pw  9798  r1pwALT  9799  elixx3g  13319  elfz2  13475  bifald  38081  areaquad  43205