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 ) )