Theorem sbcabel 3084

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 2785	. 2 |- ( A e. V -> A e. _V )
2		sbcexg 3057	. . . 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 3046	. . . . . 6 |- ( A e. _V -> ( [. A / x ]. ( w = { y | ph } /\ w e. B ) <-> ( [. A / x ]. w = { y | ph } /\ [. A / x ]. w e. B ) ) )
4		sbcalg 3055	. . . . . . . . 9 |- ( A e. _V -> ( [. A / x ]. A. y ( y e. w <-> ph ) <-> A. y [. A / x ]. ( y e. w <-> ph ) ) )
5		sbcbig 3049	. . . . . . . . . . 11 |- ( A e. _V -> ( [. A / x ]. ( y e. w <-> ph ) <-> ( [. A / x ]. y e. w <-> [. A / x ]. ph ) ) )
6		sbcg 3072	. . . . . . . . . . . 12 |- ( A e. _V -> ( [. A / x ]. y e. w <-> y e. w ) )
7	6	bibi1d 233	. . . . . . . . . . 11 |- ( A e. _V -> ( ( [. A / x ]. y e. w <-> [. A / x ]. ph ) <-> ( y e. w <-> [. A / x ]. ph ) ) )
8	5, 7	bitrd 188	. . . . . . . . . 10 |- ( A e. _V -> ( [. A / x ]. ( y e. w <-> ph ) <-> ( y e. w <-> [. A / x ]. ph ) ) )
9	8	albidv 1848	. . . . . . . . 9 |- ( A e. _V -> ( A. y [. A / x ]. ( y e. w <-> ph ) <-> A. y ( y e. w <-> [. A / x ]. ph ) ) )
10	4, 9	bitrd 188	. . . . . . . 8 |- ( A e. _V -> ( [. A / x ]. A. y ( y e. w <-> ph ) <-> A. y ( y e. w <-> [. A / x ]. ph ) ) )
11		abeq2 2315	. . . . . . . . 9 |- ( w = { y | ph } <-> A. y ( y e. w <-> ph ) )
12	11	sbcbii 3062	. . . . . . . 8 |- ( [. A / x ]. w = { y | ph } <-> [. A / x ]. A. y ( y e. w <-> ph ) )
13		abeq2 2315	. . . . . . . 8 |- ( w = { y | [. A / x ]. ph } <-> A. y ( y e. w <-> [. A / x ]. ph ) )
14	10, 12, 13	3bitr4g 223	. . . . . . 7 |- ( A e. _V -> ( [. A / x ]. w = { y | ph } <-> w = { y | [. A / x ]. ph } ) )
15		sbcabel.1	. . . . . . . . 9 |- F/_ x B
16	15	nfcri 2343	. . . . . . . 8 |- F/ x w e. B
17	16	sbcgf 3070	. . . . . . 7 |- ( A e. _V -> ( [. A / x ]. w e. B <-> w e. B ) )
18	14, 17	anbi12d 473	. . . . . 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 188	. . . . 5 |- ( A e. _V -> ( [. A / x ]. ( w = { y | ph } /\ w e. B ) <-> ( w = { y | [. A / x ]. ph } /\ w e. B ) ) )
20	19	exbidv 1849	. . . 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 188	. . 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 2202	. . . 4 |- ( { y | ph } e. B <-> E. w ( w = { y | ph } /\ w e. B ) )
23	22	sbcbii 3062	. . 3 |- ( [. A / x ]. { y | ph } e. B <-> [. A / x ]. E. w ( w = { y | ph } /\ w e. B ) )
24		df-clel 2202	. . 3 |- ( { y | [. A / x ]. ph } e. B <-> E. w ( w = { y | [. A / x ]. ph } /\ w e. B ) )
25	21, 23, 24	3bitr4g 223	. 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 104   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2177   {cab 2192   F/_wnfc 2336   _Vcvv 2773   [.wsbc 3002

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3003

This theorem is referenced by:  csbexga  4180