Theorem sbcreu 3344

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 3277	. 2 |- ( [. A / x ]. E! y e. B ph -> A e. _V )
2		reurex 3009	. . 3 |- ( E! y e. B [. A / x ]. ph -> E. y e. B [. A / x ]. ph )
3		sbcex 3277	. . . 4 |- ( [. A / x ]. ph -> A e. _V )
4	3	rexlimivw 2876	. . 3 |- ( E. y e. B [. A / x ]. ph -> A e. _V )
5	2, 4	syl 17	. 2 |- ( E! y e. B [. A / x ]. ph -> A e. _V )
6		dfsbcq2 3270	. . 3 |- ( z = A -> ( [ z / x ] E! y e. B ph <-> [. A / x ]. E! y e. B ph ) )
7		dfsbcq2 3270	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
8	7	reubidv 2975	. . 3 |- ( z = A -> ( E! y e. B [ z / x ] ph <-> E! y e. B [. A / x ]. ph ) )
9		nfcv 2592	. . . . 5 |- F/_ x B
10		nfs1v 2266	. . . . 5 |- F/ x [ z / x ] ph
11	9, 10	nfreu 2965	. . . 4 |- F/ x E! y e. B [ z / x ] ph
12		sbequ12 2083	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
13	12	reubidv 2975	. . . 4 |- ( x = z -> ( E! y e. B ph <-> E! y e. B [ z / x ] ph ) )
14	11, 13	sbie 2237	. . 3 |- ( [ z / x ] E! y e. B ph <-> E! y e. B [ z / x ] ph )
15	6, 8, 14	vtoclbg 3108	. 2 |- ( A e. _V -> ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) )
16	1, 5, 15	pm5.21nii 355	1 |- ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph )

Syntax hints:   <-> wb 188   = wceq 1444   [wsb 1797   e. wcel 1887   E.wrex 2738   E!wreu 2739   _Vcvv 3045   [.wsbc 3267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-v 3047  df-sbc 3268

This theorem is referenced by: (None)