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Theorem axc16g 1945
Description: Generalization of axc16 1946. Use the latter when sufficient. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2004, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.)
Assertion
Ref Expression
axc16g  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem axc16g
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 aevlem1 1944 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  w )
2 ax-5 1709 . 2  |-  ( ph  ->  A. w ph )
3 axc112 1942 . 2  |-  ( A. z  z  =  w  ->  ( A. w ph  ->  A. z ph )
)
41, 2, 3syl2im 38 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-12 1859
This theorem depends on definitions:  df-bi 185  df-ex 1618
This theorem is referenced by:  axc16  1946  ax16gb  1947  aev  1948  aevOLD  2066  ax16nfALT  2069
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