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Theorem dveeq1 1940
Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
dveeq1  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
Distinct variable group:    x, z

Proof of Theorem dveeq1
StepHypRef Expression
1 dveeq2 1740 . 2  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
2 equcom 1637 . 2  |-  ( z  =  y  <->  y  =  z )
32albii 1402 . 2  |-  ( A. x  z  =  y  <->  A. x  y  =  z )
41, 2, 33imtr3g 202 1  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690
This theorem is referenced by:  sbal2  1943
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