Theorem axc9 2105

Description: Derive set.mm's original ax-c9 32431 from the shorter ax-13 2057. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.)

Assertion

Ref	Expression
axc9	|- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )

Proof of Theorem axc9

Step	Hyp	Ref	Expression
1		nfeqf 2104	. . 3 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )
2	1	nfrd 1930	. 2 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> A. z x = y ) )
3	2	ex 435	1 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   A.wal 1435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662

This theorem is referenced by:  ax13  2106  hbae  2114  axi12  2399  axbnd  2400  axext4dist  30454  bj-hbaeb2  31390  wl-aleq  31832  ax12eq  32481  ax12indalem  32485