Theorem bicom 191

Description: Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
bicom	⊢ ((φ ↔ ψ) ↔ (ψ ↔ φ))

Proof of Theorem bicom

Step	Hyp	Ref	Expression
1		bicom1 190	. 2 ⊢ ((φ ↔ ψ) → (ψ ↔ φ))
2		bicom1 190	. 2 ⊢ ((ψ ↔ φ) → (φ ↔ ψ))
3	1, 2	impbii 180	1 ⊢ ((φ ↔ ψ) ↔ (ψ ↔ φ))

Syntax hints:   ↔ wb 176

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 177

This theorem is referenced by:  bicomd  192  bibi1i  305  bibi1d  310  con2bi  318  ibibr  332  bibif  335  nbbn  347  pm5.17  858  biluk  899  bigolden  901  xorcom  1307  falbitru  1352  3impexpbicom  1367  mtp-xorOLD  1537  exists1  2293  eqcom  2355  abeq1  2459  ssequn1  3433  isocnv  5491