Theorem 2falsed 714

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 628	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		2falsed.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
4	3	pm2.21d 628	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
5	2, 4	impbid 129	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-in2 624

This proof depends on definitions:  df-bi 117

This theorem is used by:  pm5.21ni  715  bianfd  961  abvor0dc  3545  nn0eln0  4767  nntri3  6770  fin0  7189  2omap  7318  omp1eomlem  7434  ctssdccl  7451  ismkvnex  7495  xrlttri3  10201  nltpnft  10218  ngtmnft  10221  xrrebnd  10223  xltadd1  10280  xposdif  10286  xleaddadd  10291  xqltnle  10704  hashnncl  11236  zfz1isolemiso  11293  mod2eq1n2dvds  12648  m1exp1  12670  bitsmod  12725  pceq0  13103