Theorem 2falsed 703

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 620	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		2falsed.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
4	3	pm2.21d 620	. 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 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm5.21ni  704  bianfd  950  abvor0dc  3474  nn0eln0  4656  nntri3  6555  fin0  6946  omp1eomlem  7160  ctssdccl  7177  ismkvnex  7221  xrlttri3  9872  nltpnft  9889  ngtmnft  9892  xrrebnd  9894  xltadd1  9951  xposdif  9957  xleaddadd  9962  xqltnle  10357  hashnncl  10887  zfz1isolemiso  10931  mod2eq1n2dvds  12044  m1exp1  12066  pceq0  12491