Theorem equequ2 1646

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 1643	. 2 |- ( x = y -> ( z = x -> z = y ) )
2		equtrr 1643	. . 3 |- ( y = x -> ( z = y -> z = x ) )
3	2	equcoms 1641	. 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 1383  ax-ie2 1428  ax-8 1440  ax-17 1464  ax-i9 1468

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  ax11v2  1748  ax11v  1755  ax11ev  1756  equs5or  1758  eujust  1950  euf  1953  mo23  1989  eleq1w  2148  disjiun  3840  iotaval  4991  dffun4f  5031  dff13f  5549  supmoti  6688  isoti  6702