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Metamath Proof Explorer

Theorem eqcom

Description: Commutative law for class equality. Theorem 6.5 of Quine p. 41. (Contributed by NM, 26-May-1993) (Proof shortened by Wolf Lammen, 19-Nov-2019)

Ref Expression
Assertion eqcom 
|- ( A = B <-> B = A )

Proof

Step Hyp Ref Expression
1 id 
 |-  ( A = B -> A = B )
2 1 eqcomd 
 |-  ( A = B -> B = A )
3 id 
 |-  ( B = A -> B = A )
4 3 eqcomd 
 |-  ( B = A -> A = B )
5 2 4 impbii 
 |-  ( A = B <-> B = A )