Theorem equequ1 1760

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 1553	. 2 |- ( x = y -> ( x = z -> y = z ) )
2		equtr 1757	. 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 1498  ax-ie2 1543  ax-8 1553  ax-17 1575  ax-i9 1579

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  equveli  1808  drsb1  1848  equsb3lem  2006  euequ1  2178  axext3  2217  cbvreuvw  2786  reu6  3009  reu7  3015  reu8nf  3127  disjiun  4109  cbviota  5322  dff13f  5949  poxp  6441  dcdifsnid  6750  modom  7074  supmoti  7297  isoti  7311  nninfwlpoim  7483  exmidontriimlem3  7543  exmidontriim  7545  netap  7584  fsum2dlemstep  12145  ennnfonelemr  13258  ctinf  13265  reap0  16969