Theorem sbcabel 3124

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 2868	. 2 |- (A e. V -> A e. _V)
2		sbcexg 3097	. . . 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 3090	. . . . . 6 |- (A e. _V -> ( [.A / x ].(w = {y | ph} /\ w e. B) <-> ( [.A / x ].w = {y | ph} /\ [.A / x ].w e. B)))
4		abeq2 2459	. . . . . . . . . 10 |- (w = {y | ph} <-> A.y(y e. w <-> ph))
5	4	sbcbii 3102	. . . . . . . . 9 |- ( [.A / x ].w = {y | ph} <-> [.A / x ].A.y(y e. w <-> ph))
6		sbcalg 3095	. . . . . . . . . 10 |- (A e. _V -> ( [.A / x ].A.y(y e. w <-> ph) <-> A.y [.A / x ].(y e. w <-> ph)))
7		sbcbig 3093	. . . . . . . . . . . 12 |- (A e. _V -> ( [.A / x ].(y e. w <-> ph) <-> ( [.A / x ].y e. w <-> [.A / x ].ph)))
8		sbcg 3112	. . . . . . . . . . . . 13 |- (A e. _V -> ( [.A / x ].y e. w <-> y e. w))
9	8	bibi1d 310	. . . . . . . . . . . 12 |- (A e. _V -> (( [.A / x ].y e. w <-> [.A / x ].ph) <-> (y e. w <-> [.A / x ].ph)))
10	7, 9	bitrd 244	. . . . . . . . . . 11 |- (A e. _V -> ( [.A / x ].(y e. w <-> ph) <-> (y e. w <-> [.A / x ].ph)))
11	10	albidv 1625	. . . . . . . . . 10 |- (A e. _V -> (A.y [.A / x ].(y e. w <-> ph) <-> A.y(y e. w <-> [.A / x ].ph)))
12	6, 11	bitrd 244	. . . . . . . . 9 |- (A e. _V -> ( [.A / x ].A.y(y e. w <-> ph) <-> A.y(y e. w <-> [.A / x ].ph)))
13	5, 12	syl5bb 248	. . . . . . . 8 |- (A e. _V -> ( [.A / x ].w = {y | ph} <-> A.y(y e. w <-> [.A / x ].ph)))
14		abeq2 2459	. . . . . . . 8 |- (w = {y | [.A / x ].ph} <-> A.y(y e. w <-> [.A / x ].ph))
15	13, 14	syl6bbr 254	. . . . . . 7 |- (A e. _V -> ( [.A / x ].w = {y | ph} <-> w = {y | [.A / x ].ph}))
16		sbcabel.1	. . . . . . . . 9 |- F/_xB
17	16	nfcri 2484	. . . . . . . 8 |- F/x w e. B
18	17	sbcgf 3110	. . . . . . 7 |- (A e. _V -> ( [.A / x ].w e. B <-> w e. B))
19	15, 18	anbi12d 691	. . . . . 6 |- (A e. _V -> (( [.A / x ].w = {y | ph} /\ [.A / x ].w e. B) <-> (w = {y | [.A / x ].ph} /\ w e. B)))
20	3, 19	bitrd 244	. . . . 5 |- (A e. _V -> ( [.A / x ].(w = {y | ph} /\ w e. B) <-> (w = {y | [.A / x ].ph} /\ w e. B)))
21	20	exbidv 1626	. . . 4 |- (A e. _V -> (E.w [.A / x ].(w = {y | ph} /\ w e. B) <-> E.w(w = {y | [.A / x ].ph} /\ w e. B)))
22	2, 21	bitrd 244	. . 3 |- (A e. _V -> ( [.A / x ].E.w(w = {y | ph} /\ w e. B) <-> E.w(w = {y | [.A / x ].ph} /\ w e. B)))
23		df-clel 2349	. . . 4 |- ({y | ph} e. B <-> E.w(w = {y | ph} /\ w e. B))
24	23	sbcbii 3102	. . 3 |- ( [.A / x ].{y | ph} e. B <-> [.A / x ].E.w(w = {y | ph} /\ w e. B))
25		df-clel 2349	. . 3 |- ({y | [.A / x ].ph} e. B <-> E.w(w = {y | [.A / x ].ph} /\ w e. B))
26	22, 24, 25	3bitr4g 279	. 2 |- (A e. _V -> ( [.A / x ].{y | ph} e. B <-> {y | [.A / x ].ph} e. B))
27	1, 26	syl 15	1 |- (A e. V -> ( [.A / x ].{y | ph} e. B <-> {y | [.A / x ].ph} e. B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339   F/_wnfc 2477  _Vcvv 2860   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  csbexg  3147