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  2004  euequ1  2176  axext3  2215  cbvreuvw  2784  reu6  3006  reu7  3012  reu8nf  3124  disjiun  4104  cbviota  5317  dff13f  5943  poxp  6428  dcdifsnid  6737  modom  7061  supmoti  7284  isoti  7298  nninfwlpoim  7470  exmidontriimlem3  7530  exmidontriim  7532  netap  7568  fsum2dlemstep  12120  ennnfonelemr  13174  ctinf  13181  reap0  16843