Theorem bdbm1.3ii 7309

Description: Bounded version of bm1.3ii 3848. (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 1579	. . . . . . . 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 1683	. . . . . 6 |- (x = z -> (A.y(ph -> y e. x) <-> A.y(ph -> y e. z)))
5	4	cbvexv 1773	. . . . 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	bdsep2 7304	. . . 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 1561	. 2 |- E.z(A.y(ph -> y e. z) /\ E.xA.y(y e. x <-> (y e. z /\ ph)))
11		19.42v 1764	. . . 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 856	. . . . . 6 |- (((ph -> y e. z) /\ (y e. x <-> (y e. z /\ ph))) -> (y e. x <-> ph))
13	12	alanimi 1324	. . . . 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 1469	. . . 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 1467	. 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 1224  E.wex 1358  BOUNDED wbd 7231

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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-ext 2000  ax-bdsep 7303

This theorem depends on definitions:  df-bi 110  df-nf 1326  df-cleq 2011  df-clel 2014

This theorem is referenced by:  bj-zfpair2  7325  bj-axun2  7330  bj-uniex2  7331