Theorem sbcabel 3123

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

Hypothesis

Ref	Expression
sbcabel.1	⊢ ℲxB

Assertion

Ref	Expression
sbcabel	⊢ (A ∈ V → ([̣A / x]̣{y ∣ φ} ∈ B ↔ {y ∣ [̣A / x]̣φ} ∈ B))

Distinct variable groups:   y,A   x,y

Allowed substitution hints:   φ(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 2867	. 2 ⊢ (A ∈ V → A ∈ V)
2		sbcexg 3096	. . . 4 ⊢ (A ∈ V → ([̣A / x]̣∃w(w = {y ∣ φ} ∧ w ∈ B) ↔ ∃w[̣A / x]̣(w = {y ∣ φ} ∧ w ∈ B)))
3		sbcang 3089	. . . . . 6 ⊢ (A ∈ V → ([̣A / x]̣(w = {y ∣ φ} ∧ w ∈ B) ↔ ([̣A / x]̣w = {y ∣ φ} ∧ [̣A / x]̣w ∈ B)))
4		abeq2 2458	. . . . . . . . . 10 ⊢ (w = {y ∣ φ} ↔ ∀y(y ∈ w ↔ φ))
5	4	sbcbii 3101	. . . . . . . . 9 ⊢ ([̣A / x]̣w = {y ∣ φ} ↔ [̣A / x]̣∀y(y ∈ w ↔ φ))
6		sbcalg 3094	. . . . . . . . . 10 ⊢ (A ∈ V → ([̣A / x]̣∀y(y ∈ w ↔ φ) ↔ ∀y[̣A / x]̣(y ∈ w ↔ φ)))
7		sbcbig 3092	. . . . . . . . . . . 12 ⊢ (A ∈ V → ([̣A / x]̣(y ∈ w ↔ φ) ↔ ([̣A / x]̣y ∈ w ↔ [̣A / x]̣φ)))
8		sbcg 3111	. . . . . . . . . . . . 13 ⊢ (A ∈ V → ([̣A / x]̣y ∈ w ↔ y ∈ w))
9	8	bibi1d 310	. . . . . . . . . . . 12 ⊢ (A ∈ V → (([̣A / x]̣y ∈ w ↔ [̣A / x]̣φ) ↔ (y ∈ w ↔ [̣A / x]̣φ)))
10	7, 9	bitrd 244	. . . . . . . . . . 11 ⊢ (A ∈ V → ([̣A / x]̣(y ∈ w ↔ φ) ↔ (y ∈ w ↔ [̣A / x]̣φ)))
11	10	albidv 1625	. . . . . . . . . 10 ⊢ (A ∈ V → (∀y[̣A / x]̣(y ∈ w ↔ φ) ↔ ∀y(y ∈ w ↔ [̣A / x]̣φ)))
12	6, 11	bitrd 244	. . . . . . . . 9 ⊢ (A ∈ V → ([̣A / x]̣∀y(y ∈ w ↔ φ) ↔ ∀y(y ∈ w ↔ [̣A / x]̣φ)))
13	5, 12	syl5bb 248	. . . . . . . 8 ⊢ (A ∈ V → ([̣A / x]̣w = {y ∣ φ} ↔ ∀y(y ∈ w ↔ [̣A / x]̣φ)))
14		abeq2 2458	. . . . . . . 8 ⊢ (w = {y ∣ [̣A / x]̣φ} ↔ ∀y(y ∈ w ↔ [̣A / x]̣φ))
15	13, 14	syl6bbr 254	. . . . . . 7 ⊢ (A ∈ V → ([̣A / x]̣w = {y ∣ φ} ↔ w = {y ∣ [̣A / x]̣φ}))
16		sbcabel.1	. . . . . . . . 9 ⊢ ℲxB
17	16	nfcri 2483	. . . . . . . 8 ⊢ Ⅎx w ∈ B
18	17	sbcgf 3109	. . . . . . 7 ⊢ (A ∈ V → ([̣A / x]̣w ∈ B ↔ w ∈ B))
19	15, 18	anbi12d 691	. . . . . 6 ⊢ (A ∈ V → (([̣A / x]̣w = {y ∣ φ} ∧ [̣A / x]̣w ∈ B) ↔ (w = {y ∣ [̣A / x]̣φ} ∧ w ∈ B)))
20	3, 19	bitrd 244	. . . . 5 ⊢ (A ∈ V → ([̣A / x]̣(w = {y ∣ φ} ∧ w ∈ B) ↔ (w = {y ∣ [̣A / x]̣φ} ∧ w ∈ B)))
21	20	exbidv 1626	. . . 4 ⊢ (A ∈ V → (∃w[̣A / x]̣(w = {y ∣ φ} ∧ w ∈ B) ↔ ∃w(w = {y ∣ [̣A / x]̣φ} ∧ w ∈ B)))
22	2, 21	bitrd 244	. . 3 ⊢ (A ∈ V → ([̣A / x]̣∃w(w = {y ∣ φ} ∧ w ∈ B) ↔ ∃w(w = {y ∣ [̣A / x]̣φ} ∧ w ∈ B)))
23		df-clel 2349	. . . 4 ⊢ ({y ∣ φ} ∈ B ↔ ∃w(w = {y ∣ φ} ∧ w ∈ B))
24	23	sbcbii 3101	. . 3 ⊢ ([̣A / x]̣{y ∣ φ} ∈ B ↔ [̣A / x]̣∃w(w = {y ∣ φ} ∧ w ∈ B))
25		df-clel 2349	. . 3 ⊢ ({y ∣ [̣A / x]̣φ} ∈ B ↔ ∃w(w = {y ∣ [̣A / x]̣φ} ∧ w ∈ B))
26	22, 24, 25	3bitr4g 279	. 2 ⊢ (A ∈ V → ([̣A / x]̣{y ∣ φ} ∈ B ↔ {y ∣ [̣A / x]̣φ} ∈ B))
27	1, 26	syl 15	1 ⊢ (A ∈ V → ([̣A / x]̣{y ∣ φ} ∈ B ↔ {y ∣ [̣A / x]̣φ} ∈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476  Vcvv 2859  [̣wsbc 3046

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-sbc 3047

This theorem is referenced by:  csbexg  3146