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.

Step	Hyp	Ref	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 ) )
4	1, 2, 3	syl2im 39	1 |- ( A. x x = y -> ( ph -> A. z ph ) )

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