Theorem ax13lem1 2248

Description: A version of ax13v 2247 with one distinct variable restriction dropped. For convenience, y is kept on the right side of equations. The proof of ax13 2249 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
ax13lem1	|- ( -. x = y -> ( z = y -> A. x z = y ) )

Distinct variable group:   x, z

Proof of Theorem ax13lem1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equviniva 1960	. 2 |- ( z = y -> E. w ( z = w /\ y = w ) )
2		ax13v 2247	. . . . 5 |- ( -. x = y -> ( y = w -> A. x y = w ) )
3		equeucl 1951	. . . . . 6 |- ( z = w -> ( y = w -> z = y ) )
4	3	alimdv 1845	. . . . 5 |- ( z = w -> ( A. x y = w -> A. x z = y ) )
5	2, 4	syl9 77	. . . 4 |- ( -. x = y -> ( z = w -> ( y = w -> A. x z = y ) ) )
6	5	impd 447	. . 3 |- ( -. x = y -> ( ( z = w /\ y = w ) -> A. x z = y ) )
7	6	exlimdv 1861	. 2 |- ( -. x = y -> ( E. w ( z = w /\ y = w ) -> A. x z = y ) )
8	1, 7	syl5 34	1 |- ( -. x = y -> ( z = y -> A. x z = y ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704

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-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  ax13  2249  ax6e  2250  ax13lem2  2296  nfeqf2  2297  wl-19.8eqv  33309  wl-19.2reqv  33310  wl-dveeq12  33311