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  2993  reu7  2999  reu8nf  3111  disjiun  4081  cbviota  5289  dff13f  5906  poxp  6392  dcdifsnid  6667  modom  6989  supmoti  7183  isoti  7197  nninfwlpoim  7369  exmidontriimlem3  7428  exmidontriim  7430  netap  7463  fsum2dlemstep  11985  ennnfonelemr  13034  ctinf  13041  reap0  16598