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  4769  nntri3  6775  fin0  7195  2omap  7324  omp1eomlem  7440  ctssdccl  7457  ismkvnex  7501  xrlttri3  10215  nltpnft  10232  ngtmnft  10235  xrrebnd  10237  xltadd1  10294  xposdif  10300  xleaddadd  10305  xqltnle  10718  flaplt  10738  hashnncl  11255  zfz1isolemiso  11312  mod2eq1n2dvds  12670  m1exp1  12692  bitsmod  12747  pceq0  13129