Theorem dtruALT 4646

Description: Alternate proof of dtru 4608 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 3170 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 4589	. . . 4 |- ( y = (/) -> -. y = { (/) } )
2		p0ex 4604	. . . . 5 |- { (/) } e. _V
3		eqeq2 2435	. . . . . 6 |- ( x = { (/) } -> ( y = x <-> y = { (/) } ) )
4	3	notbid 295	. . . . 5 |- ( x = { (/) } -> ( -. y = x <-> -. y = { (/) } ) )
5	2, 4	spcev 3170	. . . 4 |- ( -. y = { (/) } -> E. x -. y = x )
6	1, 5	syl 17	. . 3 |- ( y = (/) -> E. x -. y = x )
7		0ex 4549	. . . 4 |- (/) e. _V
8		eqeq2 2435	. . . . 5 |- ( x = (/) -> ( y = x <-> y = (/) ) )
9	8	notbid 295	. . . 4 |- ( x = (/) -> ( -. y = x <-> -. y = (/) ) )
10	7, 9	spcev 3170	. . 3 |- ( -. y = (/) -> E. x -. y = x )
11	6, 10	pm2.61i 167	. 2 |- E. x -. y = x
12		exnal 1695	. . 3 |- ( E. x -. y = x <-> -. A. x y = x )
13		eqcom 2429	. . . 4 |- ( y = x <-> x = y )
14	13	albii 1687	. . 3 |- ( A. x y = x <-> A. x x = y )
15	12, 14	xchbinx 311	. 2 |- ( E. x -. y = x <-> -. A. x x = y )
16	11, 15	mpbi 211	1 |- -. A. x x = y

Syntax hints:   -. wn 3   A.wal 1435   = wceq 1437   E.wex 1659   (/)c0 3758   {csn 3993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-v 3080  df-dif 3436  df-in 3440  df-ss 3447  df-nul 3759  df-pw 3978  df-sn 3994

This theorem is referenced by: (None)