Theorem axnul2 2698

Description: Prove axnul 2699 directly from ax-rep 2683 without using any equality axioms (ax-9 962 thru ax-16 1206). The wff x = x substituted for ph in the ax-rep 2683 instance is arbitrary. Here, we don't need to know if x = x is true or false, only that it's not both. (Contributed by Jeff Hoffman, 4-Feb-2008.)

Assertion

Ref	Expression
axnul2	|- E.xA.y -. y e. x

Distinct variable group:   x,y

Proof of Theorem axnul2

Step	Hyp	Ref	Expression
1		ax-rep 2683	. . 3 |- (A.wE.xA.y(A.x(x = x <-> -. x = x) -> y = x) -> E.xA.y(y e. x <-> E.w(w e. z /\ A.x(x = x <-> -. x = x))))
2		pm5.19 667	. . . . . . . . 9 |- -. (x = x <-> -. x = x)
3		ax-4 970	. . . . . . . . 9 |- (A.x(x = x <-> -. x = x) -> (x = x <-> -. x = x))
4	2, 3	mto 106	. . . . . . . 8 |- -. A.x(x = x <-> -. x = x)
5	4	intnan 689	. . . . . . 7 |- -. (w e. z /\ A.x(x = x <-> -. x = x))
6	5	nex 1097	. . . . . 6 |- -. E.w(w e. z /\ A.x(x = x <-> -. x = x))
7	6	nbn 720	. . . . 5 |- (-. y e. x <-> (y e. x <-> E.w(w e. z /\ A.x(x = x <-> -. x = x))))
8	7	albii 996	. . . 4 |- (A.y -. y e. x <-> A.y(y e. x <-> E.w(w e. z /\ A.x(x = x <-> -. x = x))))
9	8	exbii 1047	. . 3 |- (E.xA.y -. y e. x <-> E.xA.y(y e. x <-> E.w(w e. z /\ A.x(x = x <-> -. x = x))))
10	1, 9	sylibr 200	. 2 |- (A.wE.xA.y(A.x(x = x <-> -. x = x) -> y = x) -> E.xA.y -. y e. x)
11		19.8a 1025	. . 3 |- (A.y(A.x(x = x <-> -. x = x) -> y = x) -> E.xA.y(A.x(x = x <-> -. x = x) -> y = x))
12	4	pm2.21i 77	. . 3 |- (A.x(x = x <-> -. x = x) -> y = x)
13	11, 12	mpg 983	. 2 |- E.xA.y(A.x(x = x <-> -. x = x) -> y = x)
14	10, 13	mpg 983	1 |- E.xA.y -. y e. x

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972  ax-rep 2683

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978

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