Theorem equequ2 1640

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 1637	. 2 |- ( x = y -> ( z = x -> z = y ) )
2		equtrr 1637	. . 3 |- ( y = x -> ( z = y -> z = x ) )
3	2	equcoms 1635	. 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 1379  ax-ie2 1424  ax-8 1436  ax-17 1460  ax-i9 1464

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  ax11v2  1742  ax11v  1749  ax11ev  1750  equs5or  1752  eujust  1944  euf  1947  mo23  1983  iotaval  4908  dffun4f  4948  dff13f  5441  supmoti  6465  isoti  6479