Theorem axc9 2302

Description: Derive set.mm's original ax-c9 34175 from the shorter ax-13 2246. (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 2301	. . 3 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )
2	1	nf5rd 2066	. 2 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> A. z x = y ) )
3	2	ex 450	1 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  ax13ALT  2305  hbae  2315  axi12  2600  axbnd  2601  axext4dist  31706  bj-ax6elem1  32651  axc11n11r  32673  bj-hbaeb2  32805  wl-aleq  33322  ax12eq  34226  ax12indalem  34230