Theorem sbcabel 3517

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 3212	. 2 |- ( A e. V -> A e. _V )
2		sbcex2 3486	. . . 4 |- ( [. A / x ]. E. w ( w = { y | ph } /\ w e. B ) <-> E. w [. A / x ]. ( w = { y | ph } /\ w e. B ) )
3		sbcan 3478	. . . . . 6 |- ( [. A / x ]. ( w = { y | ph } /\ w e. B ) <-> ( [. A / x ]. w = { y | ph } /\ [. A / x ]. w e. B ) )
4		sbcal 3485	. . . . . . . . 9 |- ( [. A / x ]. A. y ( y e. w <-> ph ) <-> A. y [. A / x ]. ( y e. w <-> ph ) )
5		sbcbig 3480	. . . . . . . . . . 11 |- ( A e. _V -> ( [. A / x ]. ( y e. w <-> ph ) <-> ( [. A / x ]. y e. w <-> [. A / x ]. ph ) ) )
6		sbcg 3503	. . . . . . . . . . . 12 |- ( A e. _V -> ( [. A / x ]. y e. w <-> y e. w ) )
7	6	bibi1d 333	. . . . . . . . . . 11 |- ( A e. _V -> ( ( [. A / x ]. y e. w <-> [. A / x ]. ph ) <-> ( y e. w <-> [. A / x ]. ph ) ) )
8	5, 7	bitrd 268	. . . . . . . . . 10 |- ( A e. _V -> ( [. A / x ]. ( y e. w <-> ph ) <-> ( y e. w <-> [. A / x ]. ph ) ) )
9	8	albidv 1849	. . . . . . . . 9 |- ( A e. _V -> ( A. y [. A / x ]. ( y e. w <-> ph ) <-> A. y ( y e. w <-> [. A / x ]. ph ) ) )
10	4, 9	syl5bb 272	. . . . . . . 8 |- ( A e. _V -> ( [. A / x ]. A. y ( y e. w <-> ph ) <-> A. y ( y e. w <-> [. A / x ]. ph ) ) )
11		abeq2 2732	. . . . . . . . 9 |- ( w = { y | ph } <-> A. y ( y e. w <-> ph ) )
12	11	sbcbii 3491	. . . . . . . 8 |- ( [. A / x ]. w = { y | ph } <-> [. A / x ]. A. y ( y e. w <-> ph ) )
13		abeq2 2732	. . . . . . . 8 |- ( w = { y | [. A / x ]. ph } <-> A. y ( y e. w <-> [. A / x ]. ph ) )
14	10, 12, 13	3bitr4g 303	. . . . . . 7 |- ( A e. _V -> ( [. A / x ]. w = { y | ph } <-> w = { y | [. A / x ]. ph } ) )
15		sbcabel.1	. . . . . . . . 9 |- F/_ x B
16	15	nfcri 2758	. . . . . . . 8 |- F/ x w e. B
17	16	sbcgf 3501	. . . . . . 7 |- ( A e. _V -> ( [. A / x ]. w e. B <-> w e. B ) )
18	14, 17	anbi12d 747	. . . . . 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	syl5bb 272	. . . . 5 |- ( A e. _V -> ( [. A / x ]. ( w = { y | ph } /\ w e. B ) <-> ( w = { y | [. A / x ]. ph } /\ w e. B ) ) )
20	19	exbidv 1850	. . . 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	syl5bb 272	. . 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 2618	. . . 4 |- ( { y | ph } e. B <-> E. w ( w = { y | ph } /\ w e. B ) )
23	22	sbcbii 3491	. . 3 |- ( [. A / x ]. { y | ph } e. B <-> [. A / x ]. E. w ( w = { y | ph } /\ w e. B ) )
24		df-clel 2618	. . 3 |- ( { y | [. A / x ]. ph } e. B <-> E. w ( w = { y | [. A / x ]. ph } /\ w e. B ) )
25	21, 23, 24	3bitr4g 303	. 2 |- ( A e. _V -> ( [. A / x ]. { y | ph } e. B <-> { y | [. A / x ]. ph } e. B ) )
26	1, 25	syl 17	1 |- ( A e. V -> ( [. A / x ]. { y | ph } e. B <-> { y | [. A / x ]. ph } e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   F/_wnfc 2751   _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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436

This theorem is referenced by:  csbexg  4792