Theorem dtruALT 4899

Description: Alternate proof of dtru 4857 which requires more axioms but is shorter and may be easier to understand.

Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that x and y be distinct. Specifically, theorem spcev 3300 requires that x must not occur in the subexpression -. y = { (/) } in step 4 nor in the subexpression -. y = (/) in step 9. The proof verifier will require that x and y be in a distinct variable group to ensure this. You can check this by deleting the $d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   x, y

Proof of Theorem dtruALT

Step	Hyp	Ref	Expression
1		0inp0 4837	. . . 4 |- ( y = (/) -> -. y = { (/) } )
2		p0ex 4853	. . . . 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-pow 4843

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-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178

This theorem is referenced by: (None)