Theorem bija 344

Description: Combine antecedents into a single bi-conditional. This inference, reminiscent of ja 153, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 229 and pm5.21im 338). (Contributed by Wolf Lammen, 13-May-2013.)

Hypotheses

Ref	Expression
bija.1	⊢ (φ → (ψ → χ))
bija.2	⊢ (¬ φ → (¬ ψ → χ))

Assertion

Ref	Expression
bija	⊢ ((φ ↔ ψ) → χ)

Proof of Theorem bija

Step	Hyp	Ref	Expression
1		bi2 189	. . 3 ⊢ ((φ ↔ ψ) → (ψ → φ))
2		bija.1	. . 3 ⊢ (φ → (ψ → χ))
3	1, 2	syli 33	. 2 ⊢ ((φ ↔ ψ) → (ψ → χ))
4		bi1 178	. . . 4 ⊢ ((φ ↔ ψ) → (φ → ψ))
5	4	con3d 125	. . 3 ⊢ ((φ ↔ ψ) → (¬ ψ → ¬ φ))
6		bija.2	. . 3 ⊢ (¬ φ → (¬ ψ → χ))
7	5, 6	syli 33	. 2 ⊢ ((φ ↔ ψ) → (¬ ψ → χ))
8	3, 7	pm2.61d 150	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: (None)