Theorem sbcrext 3038

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbcrext	|- ( F/_ y A -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )

Distinct variable groups:   x, y   x, B

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

Proof of Theorem sbcrext

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 2969	. . 3 |- ( [. A / x ]. E. y e. B ph -> A e. _V )
2	1	a1i 9	. 2 |- ( F/_ y A -> ( [. A / x ]. E. y e. B ph -> A e. _V ) )
3		nfnfc1 2320	. . 3 |- F/ y F/_ y A
4		id 19	. . . 4 |- ( F/_ y A -> F/_ y A )
5		nfcvd 2318	. . . 4 |- ( F/_ y A -> F/_ y _V )
6	4, 5	nfeld 2333	. . 3 |- ( F/_ y A -> F/ y A e. _V )
7		sbcex 2969	. . . 4 |- ( [. A / x ]. ph -> A e. _V )
8	7	2a1i 27	. . 3 |- ( F/_ y A -> ( y e. B -> ( [. A / x ]. ph -> A e. _V ) ) )
9	3, 6, 8	rexlimd2 2590	. 2 |- ( F/_ y A -> ( E. y e. B [. A / x ]. ph -> A e. _V ) )
10		sbcco 2982	. . . 4 |- ( [. A / z ]. [. z / x ]. E. y e. B ph <-> [. A / x ]. E. y e. B ph )
11		simpl 109	. . . . 5 |- ( ( A e. _V /\ F/_ y A ) -> A e. _V )
12		sbsbc 2964	. . . . . . 7 |- ( [ z / x ] E. y e. B ph <-> [. z / x ]. E. y e. B ph )
13		nfcv 2317	. . . . . . . . 9 |- F/_ x B
14		nfs1v 1937	. . . . . . . . 9 |- F/ x [ z / x ] ph
15	13, 14	nfrexxy 2514	. . . . . . . 8 |- F/ x E. y e. B [ z / x ] ph
16		sbequ12 1769	. . . . . . . . 9 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
17	16	rexbidv 2476	. . . . . . . 8 |- ( x = z -> ( E. y e. B ph <-> E. y e. B [ z / x ] ph ) )
18	15, 17	sbie 1789	. . . . . . 7 |- ( [ z / x ] E. y e. B ph <-> E. y e. B [ z / x ] ph )
19	12, 18	bitr3i 186	. . . . . 6 |- ( [. z / x ]. E. y e. B ph <-> E. y e. B [ z / x ] ph )
20		nfcvd 2318	. . . . . . . . . 10 |- ( F/_ y A -> F/_ y z )
21	20, 4	nfeqd 2332	. . . . . . . . 9 |- ( F/_ y A -> F/ y z = A )
22	3, 21	nfan1 1562	. . . . . . . 8 |- F/ y ( F/_ y A /\ z = A )
23		dfsbcq2 2963	. . . . . . . . 9 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
24	23	adantl 277	. . . . . . . 8 |- ( ( F/_ y A /\ z = A ) -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
25	22, 24	rexbid 2474	. . . . . . 7 |- ( ( F/_ y A /\ z = A ) -> ( E. y e. B [ z / x ] ph <-> E. y e. B [. A / x ]. ph ) )
26	25	adantll 476	. . . . . 6 |- ( ( ( A e. _V /\ F/_ y A ) /\ z = A ) -> ( E. y e. B [ z / x ] ph <-> E. y e. B [. A / x ]. ph ) )
27	19, 26	bitrid 192	. . . . 5 |- ( ( ( A e. _V /\ F/_ y A ) /\ z = A ) -> ( [. z / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )
28	11, 27	sbcied 2997	. . . 4 |- ( ( A e. _V /\ F/_ y A ) -> ( [. A / z ]. [. z / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )
29	10, 28	bitr3id 194	. . 3 |- ( ( A e. _V /\ F/_ y A ) -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )
30	29	expcom 116	. 2 |- ( F/_ y A -> ( A e. _V -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) ) )
31	2, 9, 30	pm5.21ndd 705	1 |- ( F/_ y A -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   [wsb 1760   e. wcel 2146   F/_wnfc 2304   E.wrex 2454   _Vcvv 2735   [.wsbc 2960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961

This theorem is referenced by:  sbcrex  3040