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Theorem dveeq1 2006
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 1802 . 2  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
2 equcom 1693 . 2  |-  ( z  =  y  <->  y  =  z )
32albii 1457 . 2  |-  ( A. x  z  =  y  <->  A. x  y  =  z )
41, 2, 33imtr3g 203 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 1340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750
This theorem is referenced by:  sbal2  2007
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