Theorem axext4dist 30440

Description: axext4 2434 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Assertion

Ref	Expression
axext4dist	|- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y <-> A. z ( z e. x <-> z e. y ) ) )

Proof of Theorem axext4dist

Step	Hyp	Ref	Expression
1		axc9 2139	. . . 4 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
2	1	imp 431	. . 3 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> A. z x = y ) )
3		nfnae 2151	. . . . 5 |- F/ z -. A. z z = x
4		nfnae 2151	. . . . 5 |- F/ z -. A. z z = y
5	3, 4	nfan 2010	. . . 4 |- F/ z ( -. A. z z = x /\ -. A. z z = y )
6		elequ2 1900	. . . . 5 |- ( x = y -> ( z e. x <-> z e. y ) )
7	6	a1i 11	. . . 4 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> ( z e. x <-> z e. y ) ) )
8	5, 7	alimd 1953	. . 3 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( A. z x = y -> A. z ( z e. x <-> z e. y ) ) )
9	2, 8	syld 45	. 2 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> A. z ( z e. x <-> z e. y ) ) )
10		axextdist 30439	. 2 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( A. z ( z e. x <-> z e. y ) -> x = y ) )
11	9, 10	impbid 194	1 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y <-> A. z ( z e. x <-> z e. y ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-cleq 2443  df-clel 2446  df-nfc 2580

This theorem is referenced by: (None)