Theorem dtruALT2 4911

Description: Alternate proof of dtru 4857 using ax-pr 4906 instead of ax-pow 4843. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dtruALT2	|- -. A. x x = y

Distinct variable group:   x, y

Proof of Theorem dtruALT2

Step	Hyp	Ref	Expression
1		0inp0 4837	. . . 4 |- ( y = (/) -> -. y = { (/) } )
2		snex 4908	. . . . 5 |- { (/) } e. _V
3		eqeq2 2633	. . . . . 6 |- ( x = { (/) } -> ( y = x <-> y = { (/) } ) )
4	3	notbid 308	. . . . 5 |- ( x = { (/) } -> ( -. y = x <-> -. y = { (/) } ) )
5	2, 4	spcev 3300	. . . 4 |- ( -. y = { (/) } -> E. x -. y = x )
6	1, 5	syl 17	. . 3 |- ( y = (/) -> E. x -. y = x )
7		0ex 4790	. . . 4 |- (/) e. _V
8		eqeq2 2633	. . . . 5 |- ( x = (/) -> ( y = x <-> y = (/) ) )
9	8	notbid 308	. . . 4 |- ( x = (/) -> ( -. y = x <-> -. y = (/) ) )
10	7, 9	spcev 3300	. . 3 |- ( -. y = (/) -> E. x -. y = x )
11	6, 10	pm2.61i 176	. 2 |- E. x -. y = x
12		exnal 1754	. . 3 |- ( E. x -. y = x <-> -. A. x y = x )
13		eqcom 2629	. . . 4 |- ( y = x <-> x = y )
14	13	albii 1747	. . 3 |- ( A. x y = x <-> A. x x = y )
15	12, 14	xchbinx 324	. 2 |- ( E. x -. y = x <-> -. A. x x = y )
16	11, 15	mpbi 220	1 |- -. A. x x = y

Syntax hints:   -. wn 3   A.wal 1481   = wceq 1483   E.wex 1704   (/)c0 3915   {csn 4177

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)