Theorem sbcabel 2833

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

Hypothesis

Ref	Expression
sbcabel.1	|- F/_xB

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 2560	. 2 |- (A e. V -> A e. _V)
2		sbcexg 2807	. . . 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 2800	. . . . . 6 |- (A e. _V -> ( [.A / x ].(w = {y | ph } /\ w e. B) <-> ( [.A / x ].w = {y | ph } /\ [.A / x ].w e. B)))
4		sbcalg 2805	. . . . . . . . 9 |- (A e. _V -> ( [.A / x ].A.y(y e. w <-> ph) <-> A.y [.A / x ].(y e. w <-> ph)))
5		sbcbig 2803	. . . . . . . . . . 11 |- (A e. _V -> ( [.A / x ].(y e. w <-> ph) <-> ( [.A / x ].y e. w <-> [.A / x ].ph)))
6		sbcg 2821	. . . . . . . . . . . 12 |- (A e. _V -> ( [.A / x ].y e. w <-> y e. w))
7	6	bibi1d 222	. . . . . . . . . . 11 |- (A e. _V -> (( [.A / x ].y e. w <-> [.A / x ].ph) <-> (y e. w <-> [.A / x ].ph)))
8	5, 7	bitrd 177	. . . . . . . . . 10 |- (A e. _V -> ( [.A / x ].(y e. w <-> ph) <-> (y e. w <-> [.A / x ].ph)))
9	8	albidv 1702	. . . . . . . . 9 |- (A e. _V -> (A.y [.A / x ].(y e. w <-> ph) <-> A.y(y e. w <-> [.A / x ].ph)))
10	4, 9	bitrd 177	. . . . . . . 8 |- (A e. _V -> ( [.A / x ].A.y(y e. w <-> ph) <-> A.y(y e. w <-> [.A / x ].ph)))
11		abeq2 2143	. . . . . . . . 9 |- (w = {y | ph } <-> A.y(y e. w <-> ph))
12	11	sbcbii 2812	. . . . . . . 8 |- ( [.A / x ].w = {y | ph } <-> [.A / x ].A.y(y e. w <-> ph))
13		abeq2 2143	. . . . . . . 8 |- (w = {y | [.A / x ].ph } <-> A.y(y e. w <-> [.A / x ].ph))
14	10, 12, 13	3bitr4g 212	. . . . . . 7 |- (A e. _V -> ( [.A / x ].w = {y | ph } <-> w = {y | [.A / x ].ph }))
15		sbcabel.1	. . . . . . . . 9 |- F/_xB
16	15	nfcri 2169	. . . . . . . 8 |- F/x w e. B
17	16	sbcgf 2819	. . . . . . 7 |- (A e. _V -> ( [.A / x ].w e. B <-> w e. B))
18	14, 17	anbi12d 442	. . . . . 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 177	. . . . 5 |- (A e. _V -> ( [.A / x ].(w = {y | ph } /\ w e. B) <-> (w = {y | [.A / x ].ph } /\ w e. B)))
20	19	exbidv 1703	. . . 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 177	. . 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 2033	. . . 4 |- ( {y | ph } e. B <-> E.w(w = {y | ph } /\ w e. B))
23	22	sbcbii 2812	. . 3 |- ( [.A / x ]. {y | ph } e. B <-> [.A / x ].E.w(w = {y | ph } /\ w e. B))
24		df-clel 2033	. . 3 |- ( {y | [.A / x ].ph } e. B <-> E.w(w = {y | [.A / x ].ph } /\ w e. B))
25	21, 23, 24	3bitr4g 212	. 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 97   <-> wb 98  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390   {cab 2023   F/_wnfc 2162   _Vcvv 2551   [.wsbc 2758

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759

This theorem is referenced by:  csbexga  3876