Theorem sbcabel 3036

Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)

Hypothesis

Ref	Expression
sbcabel.1	|- F/_ x B

Assertion

Ref	Expression
sbcabel	|- ( A e. V -> ( [. A / x ]. { y | ph } e. B <-> { y | [. A / x ]. ph } e. B ) )

Distinct variable groups:   y, A   x, y

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

Proof of Theorem sbcabel

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2741	. 2 |- ( A e. V -> A e. _V )
2		sbcexg 3009	. . . 4 |- ( A e. _V -> ( [. A / x ]. E. w ( w = { y | ph } /\ w e. B ) <-> E. w [. A / x ]. ( w = { y | ph } /\ w e. B ) ) )
3		sbcang 2998	. . . . . 6 |- ( A e. _V -> ( [. A / x ]. ( w = { y | ph } /\ w e. B ) <-> ( [. A / x ]. w = { y | ph } /\ [. A / x ]. w e. B ) ) )
4		sbcalg 3007	. . . . . . . . 9 |- ( A e. _V -> ( [. A / x ]. A. y ( y e. w <-> ph ) <-> A. y [. A / x ]. ( y e. w <-> ph ) ) )
5		sbcbig 3001	. . . . . . . . . . 11 |- ( A e. _V -> ( [. A / x ]. ( y e. w <-> ph ) <-> ( [. A / x ]. y e. w <-> [. A / x ]. ph ) ) )
6		sbcg 3024	. . . . . . . . . . . 12 |- ( A e. _V -> ( [. A / x ]. y e. w <-> y e. w ) )
7	6	bibi1d 232	. . . . . . . . . . 11 |- ( A e. _V -> ( ( [. A / x ]. y e. w <-> [. A / x ]. ph ) <-> ( y e. w <-> [. A / x ]. ph ) ) )
8	5, 7	bitrd 187	. . . . . . . . . 10 |- ( A e. _V -> ( [. A / x ]. ( y e. w <-> ph ) <-> ( y e. w <-> [. A / x ]. ph ) ) )
9	8	albidv 1817	. . . . . . . . 9 |- ( A e. _V -> ( A. y [. A / x ]. ( y e. w <-> ph ) <-> A. y ( y e. w <-> [. A / x ]. ph ) ) )
10	4, 9	bitrd 187	. . . . . . . 8 |- ( A e. _V -> ( [. A / x ]. A. y ( y e. w <-> ph ) <-> A. y ( y e. w <-> [. A / x ]. ph ) ) )
11		abeq2 2279	. . . . . . . . 9 |- ( w = { y | ph } <-> A. y ( y e. w <-> ph ) )
12	11	sbcbii 3014	. . . . . . . 8 |- ( [. A / x ]. w = { y | ph } <-> [. A / x ]. A. y ( y e. w <-> ph ) )
13		abeq2 2279	. . . . . . . 8 |- ( w = { y | [. A / x ]. ph } <-> A. y ( y e. w <-> [. A / x ]. ph ) )
14	10, 12, 13	3bitr4g 222	. . . . . . 7 |- ( A e. _V -> ( [. A / x ]. w = { y | ph } <-> w = { y | [. A / x ]. ph } ) )
15		sbcabel.1	. . . . . . . . 9 |- F/_ x B
16	15	nfcri 2306	. . . . . . . 8 |- F/ x w e. B
17	16	sbcgf 3022	. . . . . . 7 |- ( A e. _V -> ( [. A / x ]. w e. B <-> w e. B ) )
18	14, 17	anbi12d 470	. . . . . 6 |- ( A e. _V -> ( ( [. A / x ]. w = { y | ph } /\ [. A / x ]. w e. B ) <-> ( w = { y | [. A / x ]. ph } /\ w e. B ) ) )
19	3, 18	bitrd 187	. . . . 5 |- ( A e. _V -> ( [. A / x ]. ( w = { y | ph } /\ w e. B ) <-> ( w = { y | [. A / x ]. ph } /\ w e. B ) ) )
20	19	exbidv 1818	. . . 4 |- ( A e. _V -> ( E. w [. A / x ]. ( w = { y | ph } /\ w e. B ) <-> E. w ( w = { y | [. A / x ]. ph } /\ w e. B ) ) )
21	2, 20	bitrd 187	. . 3 |- ( A e. _V -> ( [. A / x ]. E. w ( w = { y | ph } /\ w e. B ) <-> E. w ( w = { y | [. A / x ]. ph } /\ w e. B ) ) )
22		df-clel 2166	. . . 4 |- ( { y | ph } e. B <-> E. w ( w = { y | ph } /\ w e. B ) )
23	22	sbcbii 3014	. . 3 |- ( [. A / x ]. { y | ph } e. B <-> [. A / x ]. E. w ( w = { y | ph } /\ w e. B ) )
24		df-clel 2166	. . 3 |- ( { y | [. A / x ]. ph } e. B <-> E. w ( w = { y | [. A / x ]. ph } /\ w e. B ) )
25	21, 23, 24	3bitr4g 222	. 2 |- ( A e. _V -> ( [. A / x ]. { y | ph } e. B <-> { y | [. A / x ]. ph } e. B ) )
26	1, 25	syl 14	1 |- ( A e. V -> ( [. A / x ]. { y | ph } e. B <-> { y | [. A / x ]. ph } e. B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   F/_wnfc 2299   _Vcvv 2730   [.wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956

This theorem is referenced by:  csbexga  4117