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 3334	. 2 |- ( [. A / x ]. E! y e. B ph -> A e. _V )
2		reurex 3071	. . 3 |- ( E! y e. B [. A / x ]. ph -> E. y e. B [. A / x ]. ph )
3		sbcex 3334	. . . 4 |- ( [. A / x ]. ph -> A e. _V )
4	3	rexlimivw 2945	. . 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 3327	. . 3 |- ( z = A -> ( [ z / x ] E! y e. B ph <-> [. A / x ]. E! y e. B ph ) )
7		dfsbcq2 3327	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
8	7	reubidv 3039	. . 3 |- ( z = A -> ( E! y e. B [ z / x ] ph <-> E! y e. B [. A / x ]. ph ) )
9		nfcv 2622	. . . . 5 |- F/_ x B
10		nfs1v 2157	. . . . 5 |- F/ x [ z / x ] ph
11	9, 10	nfreu 3029	. . . 4 |- F/ x E! y e. B [ z / x ] ph
12		sbequ12 1954	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
13	12	reubidv 3039	. . . 4 |- ( x = z -> ( E! y e. B ph <-> E! y e. B [ z / x ] ph ) )
14	11, 13	sbie 2116	. . 3 |- ( [ z / x ] E! y e. B ph <-> E! y e. B [ z / x ] ph )
15	6, 8, 14	vtoclbg 3165	. 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 1374   [wsb 1706   e. wcel 1762   E.wrex 2808   E!wreu 2809   _Vcvv 3106   [.wsbc 3324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-v 3108  df-sbc 3325

This theorem is referenced by: (None)