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Theorem a12stdy2 29749
Description: Part of a study related to ax12o 1887. The consequent is quantified with a different variable. There are no distinct variable restrictions. (Contributed by NM, 14-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a12stdy2  |-  ( A. z ( z  =  x  /\  x  =  y )  ->  A. y 
y  =  x )

Proof of Theorem a12stdy2
StepHypRef Expression
1 19.26 1583 . 2  |-  ( A. z ( z  =  x  /\  x  =  y )  <->  ( A. z  z  =  x  /\  A. z  x  =  y ) )
2 ax10o 1905 . . . 4  |-  ( A. z  z  =  x  ->  ( A. z  x  =  y  ->  A. x  x  =  y )
)
3 aecom 1899 . . . 4  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
42, 3syl6 29 . . 3  |-  ( A. z  z  =  x  ->  ( A. z  x  =  y  ->  A. y 
y  =  x ) )
54imp 418 . 2  |-  ( ( A. z  z  =  x  /\  A. z  x  =  y )  ->  A. y  y  =  x )
61, 5sylbi 187 1  |-  ( A. z ( z  =  x  /\  x  =  y )  ->  A. y 
y  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530
This theorem is referenced by:  a12stdy3  29750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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