Theorem equequ2 1639

Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
equequ2	|- ( x = y -> ( z = x <-> z = y ) )

Proof of Theorem equequ2

Step	Hyp	Ref	Expression
1		equtrr 1636	. 2 |- ( x = y -> ( z = x -> z = y ) )
2		equtrr 1636	. . 3 |- ( y = x -> ( z = y -> z = x ) )
3	2	equcoms 1634	. 2 |- ( x = y -> ( z = y -> z = x ) )
4	1, 3	impbid 127	1 |- ( x = y -> ( z = x <-> z = y ) )

Syntax hints:   -> wi 4   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1378  ax-ie2 1423  ax-8 1435  ax-17 1459  ax-i9 1463

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  ax11v2  1741  ax11v  1748  ax11ev  1749  equs5or  1751  eujust  1943  euf  1946  mo23  1982  iotaval  4898  dffun4f  4938  dff13f  5430  supmoti  6406  isoti  6420