Theorem bicom1 223

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 222	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2		biimp 217	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
3	1, 2	impbid 214	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 208

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 209

This theorem is referenced by:  bicom  224  bicomi  226  nanass  1518  rexsng  4611  axpow3  5300  xpord3inddlem  8098  nummin  35289  bj-axreprepsep  37443  bicomdALT  43130  frege55aid  44324  frege55lem2a  44326  bisaiaisb  47390  confun4  47419  confun5  47420  ichnfim  47953