Theorem 2falsed 618

Description: Two falsehoods are equivalent (deduction rule). (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 549	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		2falsed.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
4	3	pm2.21d 549	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
5	2, 4	impbid 120	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 100  ax-ia3 101  ax-in2 545

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  pm5.21ni  619  bianfd  855  abvor0dc  3242  nn0eln0  4341  nntri3  6075  fin0  6342  xrlttri3  8718  nltpnft  8730  ngtmnft  8731  xrrebnd  8732