Theorem 2falsed 697

Description: Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.)

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	1	pm2.21d 614	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		2falsed.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
4	3	pm2.21d 614	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
5	2, 4	impbid 128	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106  ax-ia3 107  ax-in2 610

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm5.21ni  698  bianfd  943  abvor0dc  3437  nn0eln0  4602  nntri3  6473  fin0  6859  omp1eomlem  7067  ctssdccl  7084  ismkvnex  7127  xrlttri3  9741  nltpnft  9758  ngtmnft  9761  xrrebnd  9763  xltadd1  9820  xposdif  9826  xleaddadd  9831  hashnncl  10717  zfz1isolemiso  10761  mod2eq1n2dvds  11825  m1exp1  11847  pceq0  12262