Theorem bdbm1.3ii 9346

Description: Bounded version of bm1.3ii 3869. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdbm1.3ii.bd	|- BOUNDED ph
bdbm1.3ii.1	|- E.xA.y(ph -> y e. x)

Assertion

Ref	Expression
bdbm1.3ii	|- E.xA.y(y e. x <-> ph)

Distinct variable groups:   ph,x   x,y

Allowed substitution hint:   ph(y)

Proof of Theorem bdbm1.3ii

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdbm1.3ii.1	. . . . 5 |- E.xA.y(ph -> y e. x)
2		elequ2 1598	. . . . . . . 8 |- (x = z -> (y e. x <-> y e. z))
3	2	imbi2d 219	. . . . . . 7 |- (x = z -> ((ph -> y e. x) <-> (ph -> y e. z)))
4	3	albidv 1702	. . . . . 6 |- (x = z -> (A.y(ph -> y e. x) <-> A.y(ph -> y e. z)))
5	4	cbvexv 1792	. . . . 5 |- (E.xA.y(ph -> y e. x) <-> E.zA.y(ph -> y e. z))
6	1, 5	mpbi 133	. . . 4 |- E.zA.y(ph -> y e. z)
7		bdbm1.3ii.bd	. . . . 5 |- BOUNDED ph
8	7	bdsep1 9340	. . . 4 |- E.xA.y(y e. x <-> (y e. z /\ ph))
9	6, 8	pm3.2i 257	. . 3 |- (E.zA.y(ph -> y e. z) /\ E.xA.y(y e. x <-> (y e. z /\ ph)))
10	9	exan 1580	. 2 |- E.z(A.y(ph -> y e. z) /\ E.xA.y(y e. x <-> (y e. z /\ ph)))
11		19.42v 1783	. . . 4 |- (E.x(A.y(ph -> y e. z) /\ A.y(y e. x <-> (y e. z /\ ph))) <-> (A.y(ph -> y e. z) /\ E.xA.y(y e. x <-> (y e. z /\ ph))))
12		bimsc1 869	. . . . . 6 |- (((ph -> y e. z) /\ (y e. x <-> (y e. z /\ ph))) -> (y e. x <-> ph))
13	12	alanimi 1345	. . . . 5 |- ((A.y(ph -> y e. z) /\ A.y(y e. x <-> (y e. z /\ ph))) -> A.y(y e. x <-> ph))
14	13	eximi 1488	. . . 4 |- (E.x(A.y(ph -> y e. z) /\ A.y(y e. x <-> (y e. z /\ ph))) -> E.xA.y(y e. x <-> ph))
15	11, 14	sylbir 125	. . 3 |- ((A.y(ph -> y e. z) /\ E.xA.y(y e. x <-> (y e. z /\ ph))) -> E.xA.y(y e. x <-> ph))
16	15	exlimiv 1486	. 2 |- (E.z(A.y(ph -> y e. z) /\ E.xA.y(y e. x <-> (y e. z /\ ph))) -> E.xA.y(y e. x <-> ph))
17	10, 16	ax-mp 7	1 |- E.xA.y(y e. x <-> ph)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378  BOUNDED wbd 9267

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-bdsep 9339

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bj-zfpair2  9365  bj-axun2  9370  bj-uniex2  9371