Theorem equequ1 1764

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 1557	. 2 |- ( x = y -> ( x = z -> y = z ) )
2		equtr 1761	. 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 1502  ax-ie2 1547  ax-8 1557  ax-17 1579  ax-i9 1583

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  equveli  1812  drsb1  1852  equsb3lem  2010  euequ1  2182  axext3  2221  cbvreuvw  2792  reu6  3015  reu7  3021  reu8nf  3133  disjiun  4120  cbviota  5337  dff13f  5966  poxp  6458  dcdifsnid  6767  modom  7098  supmoti  7323  isoti  7337  nninfwlpoim  7509  exmidontriimlem3  7569  exmidontriim  7571  netap  7610  fsum2dlemstep  12179  ennnfonelemr  13292  ctinf  13299  reap0  17013