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  1540  rexsng  4642  axpow3  5339  xpord3inddlem  8146  nummin  35493  bj-axreprepsep  37740  bicomdALT  43425  frege55aid  44619  frege55lem2a  44621  bisaiaisb  47678  confun4  47707  confun5  47708  ichnfim  48241