Theorem axext4dist 31706

Description: axext4 2606 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 2302	. . . 4 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
2	1	imp 445	. . 3 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> A. z x = y ) )
3		nfnae 2318	. . . . 5 |- F/ z -. A. z z = x
4		nfnae 2318	. . . . 5 |- F/ z -. A. z z = y
5	3, 4	nfan 1828	. . . 4 |- F/ z ( -. A. z z = x /\ -. A. z z = y )
6		elequ2 2004	. . . . 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 2081	. . 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 47	. 2 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> A. z ( z e. x <-> z e. y ) ) )
10		axextdist 31705	. 2 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( A. z ( z e. x <-> z e. y ) -> x = y ) )
11	9, 10	impbid 202	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 196   /\ 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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by: (None)