Theorem axinf 1084

Description: Axiom of Infinity expressed with fewest number of different variables.

Assertion

Ref	Expression
axinf	|- E.x(y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))

Distinct variable group(s):   x,y,z

Proof of Theorem axinf

Step	Hyp	Ref	Expression
1		ax-inf 1079	. 2 |- E.x(y e. x /\ A.w(w e. x -> E.z(w e. z /\ z e. x)))
2		a13b 819	. . . . . 6 |- (w = y -> (w e. x <-> y e. x))
3		a13b 819	. . . . . . . 8 |- (w = y -> (w e. z <-> y e. z))
4	3	anbi1d 469	. . . . . . 7 |- (w = y -> ((w e. z /\ z e. x) <-> (y e. z /\ z e. x)))
5	4	biexdv 936	. . . . . 6 |- (w = y -> (E.z(w e. z /\ z e. x) <-> E.z(y e. z /\ z e. x)))
6	2, 5	imbi12d 474	. . . . 5 |- (w = y -> ((w e. x -> E.z(w e. z /\ z e. x)) <-> (y e. x -> E.z(y e. z /\ z e. x))))
7	6	cbvalv 972	. . . 4 |- (A.w(w e. x -> E.z(w e. z /\ z e. x)) <-> A.y(y e. x -> E.z(y e. z /\ z e. x)))
8	7	anbi2i 367	. . 3 |- ((y e. x /\ A.w(w e. x -> E.z(w e. z /\ z e. x))) <-> (y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
9	8	biex 733	. 2 |- (E.x(y e. x /\ A.w(w e. x -> E.z(w e. z /\ z e. x))) <-> E.x(y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
10	1, 9	mpbi 164	1 |- E.x(y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803

This theorem is referenced by:  inf4 3473  axinfndlem1 3751

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-13 804  ax-17 925  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

metamath.org