Theorem bicom1 224

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 223	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2		biimp 218	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
3	1, 2	impbid 215	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209

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

This proof depends on definitions:  df-bi 210

This theorem is used by:  bicom  225  bicomi  227  nanass  1539  rexsng  4636  axpow3  5329  xpord3inddlem  8171  nummin  35722  bj-axreprepsep  37991  bicomdALT  43677  frege55aid  44865  frege55lem2a  44867  bisaiaisb  47983  confun4  48012  confun5  48013  ichnfim  48546