Theorem sbcabel 2985

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 2692	. 2 |- ( A e. V -> A e. _V )
2		sbcexg 2958	. . . 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 2947	. . . . . 6 |- ( A e. _V -> ( [. A / x ]. ( w = { y | ph } /\ w e. B ) <-> ( [. A / x ]. w = { y | ph } /\ [. A / x ]. w e. B ) ) )
4		sbcalg 2956	. . . . . . . . 9 |- ( A e. _V -> ( [. A / x ]. A. y ( y e. w <-> ph ) <-> A. y [. A / x ]. ( y e. w <-> ph ) ) )
5		sbcbig 2950	. . . . . . . . . . 11 |- ( A e. _V -> ( [. A / x ]. ( y e. w <-> ph ) <-> ( [. A / x ]. y e. w <-> [. A / x ]. ph ) ) )
6		sbcg 2973	. . . . . . . . . . . 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 1796	. . . . . . . . 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 2246	. . . . . . . . 9 |- ( w = { y | ph } <-> A. y ( y e. w <-> ph ) )
12	11	sbcbii 2963	. . . . . . . 8 |- ( [. A / x ]. w = { y | ph } <-> [. A / x ]. A. y ( y e. w <-> ph ) )
13		abeq2 2246	. . . . . . . 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 2273	. . . . . . . 8 |- F/ x w e. B
17	16	sbcgf 2971	. . . . . . 7 |- ( A e. _V -> ( [. A / x ]. w e. B <-> w e. B ) )
18	14, 17	anbi12d 464	. . . . . 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 1797	. . . 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 2133	. . . 4 |- ( { y | ph } e. B <-> E. w ( w = { y | ph } /\ w e. B ) )
23	22	sbcbii 2963	. . 3 |- ( [. A / x ]. { y | ph } e. B <-> [. A / x ]. E. w ( w = { y | ph } /\ w e. B ) )
24		df-clel 2133	. . 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 1329   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2123   F/_wnfc 2266   _Vcvv 2681   [.wsbc 2904

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905

This theorem is referenced by:  csbexga  4051