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Theorem a16g 2049
Description: Generalization of ax16 2051. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 18-Feb-2018.)
Assertion
Ref Expression
a16g  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem a16g
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 aevlem1 2047 . 2  |-  ( A. x  x  =  y  ->  A. w  w  =  z )
2 ax-17 1627 . 2  |-  ( ph  ->  A. w ph )
3 ax10o 2039 . 2  |-  ( A. w  w  =  z  ->  ( A. w ph  ->  A. z ph )
)
41, 2, 3syl2im 37 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550
This theorem is referenced by:  ax16  2051  a16gb  2053  a16nf  2054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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