MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc16g Structured version   Visualization version   Unicode version

Theorem axc16g 2027
Description: Generalization of axc16 2028. 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 2091, 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 2026 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  w )
2 ax-5 1761 . 2  |-  ( ph  ->  A. w ph )
3 axc112 2024 . 2  |-  ( A. z  z  =  w  ->  ( A. w ph  ->  A. z ph )
)
41, 2, 3syl2im 39 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-12 1936
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1667
This theorem is referenced by:  axc16  2028  axc16gb  2029  aev  2030  axc16nfALT  2157
  Copyright terms: Public domain W3C validator