Theorem 2falsed 376

Description: Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.) (Proof shortened by Wolf Lammen, 11-Apr-2024.)

Hypotheses

Ref	Expression
2falsed.1	⊢ (𝜑 → ¬ 𝜓)
2falsed.2	⊢ (𝜑 → ¬ 𝜒)

Assertion

Ref	Expression
2falsed	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Proof of Theorem 2falsed

Step	Hyp	Ref	Expression
1		2falsed.1	. . 3 ⊢ (𝜑 → ¬ 𝜓)
2		2falsed.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
3	1, 2	2thd 264	. 2 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
4	3	con4bid 316	1 ⊢ (𝜑 → (𝜓 ↔ 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:  pm5.21ni  378  bianfd  535  sbcel12  4408  sbcne12  4412  sbcel2  4415  sbcbr  5203  csbxp  5775  smoord  8367  tfr2b  8398  axrepnd  10591  hasheq0  14327  m1exp1  16323  sadcadd  16403  stdbdxmet  24244  iccpnfcnv  24684  cxple2  26429  mirbtwnhl  28186  eupth2lem1  29726  ifnebib  32036  isoun  32178  1smat1  33070  xrge0iifcnv  33199  sgn0bi  33832  signswch  33858  fmlafvel  34662  fz0n  34992  hfext  35447  unccur  36774  ntrneiel2  43139  ntrneik4w  43153  eliin2f  44095