Theorem axc9 2154

Description: Derive set.mm's original ax-c9 32526 from the shorter ax-13 2104. (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 2152	. . 3 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )
2	1	nfrd 1973	. 2 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> A. z x = y ) )
3	2	ex 441	1 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   A.wal 1450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676

This theorem is referenced by:  ax13  2155  hbae  2164  axi12  2449  axbnd  2450  axext4dist  30518  bj-ax6elem1  31328  bj-hbaeb2  31486  wl-aleq  31938  ax12eq  32576  ax12indalem  32580