Theorem bicom1 190

Description: Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.)

Assertion

Ref	Expression
bicom1	⊢ ((φ ↔ ψ) → (ψ ↔ φ))

Proof of Theorem bicom1

Step	Hyp	Ref	Expression
1		bi2 189	. 2 ⊢ ((φ ↔ ψ) → (ψ → φ))
2		bi1 178	. 2 ⊢ ((φ ↔ ψ) → (φ → ψ))
3	1, 2	impbid 183	1 ⊢ ((φ ↔ ψ) → (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:  bicom  191  bicomi  193