Metamath Proof Explorer


Theorem axc16g

Description: Generalization of axc16 . Use the latter when sufficient. This proof only requires, on top of { ax-1 -- ax-7 }, Theorem ax12v . (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 , along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019) (Revised by BJ, 7-Jul-2021) Shorten axc11rv . (Revised by Wolf Lammen, 11-Oct-2021)

Ref Expression
Assertion axc16g
|- ( A. x x = y -> ( ph -> A. z ph ) )

Proof

Step Hyp Ref Expression
1 aevlem
 |-  ( A. x x = y -> A. z z = w )
2 ax12v
 |-  ( z = w -> ( ph -> A. z ( z = w -> ph ) ) )
3 2 sps
 |-  ( A. z z = w -> ( ph -> A. z ( z = w -> ph ) ) )
4 pm2.27
 |-  ( z = w -> ( ( z = w -> ph ) -> ph ) )
5 4 al2imi
 |-  ( A. z z = w -> ( A. z ( z = w -> ph ) -> A. z ph ) )
6 3 5 syld
 |-  ( A. z z = w -> ( ph -> A. z ph ) )
7 1 6 syl
 |-  ( A. x x = y -> ( ph -> A. z ph ) )