Theorem axinf 4632

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:   x,y,z

Proof of Theorem axinf

Step	Hyp	Ref	Expression
1		ax-inf 4631	. 2 |- E.x(y e. x /\ A.w(w e. x -> E.z(w e. z /\ z e. x)))
2		elequ1 1138	. . . . . 6 |- (w = y -> (w e. x <-> y e. x))
3		elequ1 1138	. . . . . . . 8 |- (w = y -> (w e. z <-> y e. z))
4	3	anbi1d 619	. . . . . . 7 |- (w = y -> ((w e. z /\ z e. x) <-> (y e. z /\ z e. x)))
5	4	exbidv 1281	. . . . . 6 |- (w = y -> (E.z(w e. z /\ z e. x) <-> E.z(y e. z /\ z e. x)))
6	2, 5	imbi12d 628	. . . . 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 1316	. . . 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 482	. . 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	exbii 1053	. 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 189	1 |- E.x(y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982

This theorem is referenced by:  axinf2 4633  axinfndlem1 4969

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-13 971  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-inf 4631

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983

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