Theorem 2falsed 651

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 582	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		2falsed.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
4	3	pm2.21d 582	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
5	2, 4	impbid 127	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 105  ax-ia3 106  ax-in2 578

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  pm5.21ni  652  bianfd  890  abvor0dc  3276  nn0eln0  4367  nntri3  6141  fin0  6419  xrlttri3  8948  nltpnft  8960  ngtmnft  8961  xrrebnd  8962  sizenncl  9820  mod2eq1n2dvds  10423  m1exp1  10445