Theorem equequ1 1758

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

Assertion

Ref	Expression
equequ1	|- ( x = y -> ( x = z <-> y = z ) )

Proof of Theorem equequ1

Step	Hyp	Ref	Expression
1		ax-8 1550	. 2 |- ( x = y -> ( x = z -> y = z ) )
2		equtr 1755	. 2 |- ( x = y -> ( y = z -> x = z ) )
3	1, 2	impbid 129	1 |- ( x = y -> ( x = z <-> y = z ) )

Syntax hints:   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1495  ax-ie2 1540  ax-8 1550  ax-17 1572  ax-i9 1576

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  equveli  1805  drsb1  1845  equsb3lem  2001  euequ1  2173  axext3  2212  cbvreuvw  2771  reu6  2992  reu7  2998  reu8nf  3110  disjiun  4077  cbviota  5282  dff13f  5893  poxp  6376  dcdifsnid  6648  supmoti  7156  isoti  7170  nninfwlpoim  7342  exmidontriimlem3  7401  exmidontriim  7403  netap  7436  fsum2dlemstep  11940  ennnfonelemr  12989  ctinf  12996  reap0  16385