Theorem axextdist 31705

Description: ax-ext 2602 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Assertion

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

Proof of Theorem axextdist

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfnae 2318	. . . 4 |- F/ z -. A. z z = x
2		nfnae 2318	. . . 4 |- F/ z -. A. z z = y
3	1, 2	nfan 1828	. . 3 |- F/ z ( -. A. z z = x /\ -. A. z z = y )
4		nfcvf 2788	. . . . . 6 |- ( -. A. z z = x -> F/_ z x )
5	4	adantr 481	. . . . 5 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/_ z x )
6	5	nfcrd 2771	. . . 4 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z w e. x )
7		nfcvf 2788	. . . . . 6 |- ( -. A. z z = y -> F/_ z y )
8	7	adantl 482	. . . . 5 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/_ z y )
9	8	nfcrd 2771	. . . 4 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z w e. y )
10	6, 9	nfbid 1832	. . 3 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z ( w e. x <-> w e. y ) )
11		elequ1 1997	. . . . 5 |- ( w = z -> ( w e. x <-> z e. x ) )
12		elequ1 1997	. . . . 5 |- ( w = z -> ( w e. y <-> z e. y ) )
13	11, 12	bibi12d 335	. . . 4 |- ( w = z -> ( ( w e. x <-> w e. y ) <-> ( z e. x <-> z e. y ) ) )
14	13	a1i 11	. . 3 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( w = z -> ( ( w e. x <-> w e. y ) <-> ( z e. x <-> z e. y ) ) ) )
15	3, 10, 14	cbvald 2277	. 2 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( A. w ( w e. x <-> w e. y ) <-> A. z ( z e. x <-> z e. y ) ) )
16		axext3 2604	. 2 |- ( A. w ( w e. x <-> w e. y ) -> x = y )
17	15, 16	syl6bir 244	1 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( A. z ( z e. x <-> z e. y ) -> x = y ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   F/_wnfc 2751

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:  axext4dist  31706