Theorem bicom1 220

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

Assertion

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

Proof of Theorem bicom1

Step	Hyp	Ref	Expression
1		biimpr 219	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2		biimp 214	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
3	1, 2	impbid 211	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205

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 206

This theorem is referenced by:  bicom  221  bicomi  223  nanass  1504  rexsng  4674  xpord3inddlem  8153  nummin  34708  bicomdALT  42083  frege55aid  43289  frege55lem2a  43291  bisaiaisb  46289  confun4  46318  confun5  46319  ichnfim  46798