Theorem sbcreu 3515

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 3445	. 2 |- ( [. A / x ]. E! y e. B ph -> A e. _V )
2		reurex 3160	. . 3 |- ( E! y e. B [. A / x ]. ph -> E. y e. B [. A / x ]. ph )
3		sbcex 3445	. . . 4 |- ( [. A / x ]. ph -> A e. _V )
4	3	rexlimivw 3029	. . 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 3438	. . 3 |- ( z = A -> ( [ z / x ] E! y e. B ph <-> [. A / x ]. E! y e. B ph ) )
7		dfsbcq2 3438	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
8	7	reubidv 3126	. . 3 |- ( z = A -> ( E! y e. B [ z / x ] ph <-> E! y e. B [. A / x ]. ph ) )
9		nfcv 2764	. . . . 5 |- F/_ x B
10		nfs1v 2437	. . . . 5 |- F/ x [ z / x ] ph
11	9, 10	nfreu 3114	. . . 4 |- F/ x E! y e. B [ z / x ] ph
12		sbequ12 2111	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
13	12	reubidv 3126	. . . 4 |- ( x = z -> ( E! y e. B ph <-> E! y e. B [ z / x ] ph ) )
14	11, 13	sbie 2408	. . 3 |- ( [ z / x ] E! y e. B ph <-> E! y e. B [ z / x ] ph )
15	6, 8, 14	vtoclbg 3267	. 2 |- ( A e. _V -> ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) )
16	1, 5, 15	pm5.21nii 368	1 |- ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph )

Syntax hints:   <-> wb 196   = wceq 1483   [wsb 1880   e. wcel 1990   E.wrex 2913   E!wreu 2914   _Vcvv 3200   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)