Theorem 2falsed 702

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 619	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		2falsed.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
4	3	pm2.21d 619	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
5	2, 4	impbid 129	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm5.21ni  703  bianfd  948  abvor0dc  3446  nn0eln0  4619  nntri3  6497  fin0  6884  omp1eomlem  7092  ctssdccl  7109  ismkvnex  7152  xrlttri3  9795  nltpnft  9812  ngtmnft  9815  xrrebnd  9817  xltadd1  9874  xposdif  9880  xleaddadd  9885  hashnncl  10770  zfz1isolemiso  10814  mod2eq1n2dvds  11878  m1exp1  11900  pceq0  12315