Theorem sbcreu 3411

Description: Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbcreu	|- ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph )

Distinct variable groups:   y, A   x, B   x, y

Allowed substitution hints:   ph( x, y)   A( x)   B( y)

Proof of Theorem sbcreu

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3337	. 2 |- ( [. A / x ]. E! y e. B ph -> A e. _V )
2		reurex 3074	. . 3 |- ( E! y e. B [. A / x ]. ph -> E. y e. B [. A / x ]. ph )
3		sbcex 3337	. . . 4 |- ( [. A / x ]. ph -> A e. _V )
4	3	rexlimivw 2946	. . 3 |- ( E. y e. B [. A / x ]. ph -> A e. _V )
5	2, 4	syl 16	. 2 |- ( E! y e. B [. A / x ]. ph -> A e. _V )
6		dfsbcq2 3330	. . 3 |- ( z = A -> ( [ z / x ] E! y e. B ph <-> [. A / x ]. E! y e. B ph ) )
7		dfsbcq2 3330	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
8	7	reubidv 3042	. . 3 |- ( z = A -> ( E! y e. B [ z / x ] ph <-> E! y e. B [. A / x ]. ph ) )
9		nfcv 2619	. . . . 5 |- F/_ x B
10		nfs1v 2182	. . . . 5 |- F/ x [ z / x ] ph
11	9, 10	nfreu 3032	. . . 4 |- F/ x E! y e. B [ z / x ] ph
12		sbequ12 1993	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
13	12	reubidv 3042	. . . 4 |- ( x = z -> ( E! y e. B ph <-> E! y e. B [ z / x ] ph ) )
14	11, 13	sbie 2150	. . 3 |- ( [ z / x ] E! y e. B ph <-> E! y e. B [ z / x ] ph )
15	6, 8, 14	vtoclbg 3168	. 2 |- ( A e. _V -> ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) )
16	1, 5, 15	pm5.21nii 353	1 |- ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph )

Syntax hints:   <-> wb 184   = wceq 1395   [wsb 1740   e. wcel 1819   E.wrex 2808   E!wreu 2809   _Vcvv 3109   [.wsbc 3327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-v 3111  df-sbc 3328

This theorem is referenced by: (None)