Theorem 2falsed 340

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 98	. 2 ⊢ (φ → (ψ → χ))
3		2falsed.2	. . 3 ⊢ (φ → ¬ χ)
4	3	pm2.21d 98	. 2 ⊢ (φ → (χ → ψ))
5	2, 4	impbid 183	1 ⊢ (φ → (ψ ↔ χ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  pm5.21ni  341  bianfd  892  abvor0  3568  eqfnfv  5393