Theorem ibibr 332

Description: Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)

Assertion

Ref	Expression
ibibr	⊢ ((φ → ψ) ↔ (φ → (ψ ↔ φ)))

Proof of Theorem ibibr

Step	Hyp	Ref	Expression
1		pm5.501 330	. . 3 ⊢ (φ → (ψ ↔ (φ ↔ ψ)))
2		bicom 191	. . 3 ⊢ ((φ ↔ ψ) ↔ (ψ ↔ φ))
3	1, 2	syl6bb 252	. 2 ⊢ (φ → (ψ ↔ (ψ ↔ φ)))
4	3	pm5.74i 236	1 ⊢ ((φ → ψ) ↔ (φ → (ψ ↔ φ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  tbt  333