Theorem 2falsed 707

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 622	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		2falsed.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
4	3	pm2.21d 622	. 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 618

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm5.21ni  708  bianfd  954  abvor0dc  3515  nn0eln0  4713  nntri3  6656  fin0  7060  omp1eomlem  7277  ctssdccl  7294  ismkvnex  7338  xrlttri3  10010  nltpnft  10027  ngtmnft  10030  xrrebnd  10032  xltadd1  10089  xposdif  10095  xleaddadd  10100  xqltnle  10504  hashnncl  11034  zfz1isolemiso  11079  mod2eq1n2dvds  12411  m1exp1  12433  bitsmod  12488  pceq0  12866  2omap  16472