Theorem sbcabel 1992

Description: Interchange class substitution and class abstraction.

Hypothesis

Ref	Expression
sbcabel.1	⊢ (z ∈ B → ∀x z ∈ B)

Assertion

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

Distinct variable groups:   y,A   z,B   x,y   x,z

Proof of Theorem sbcabel

Step	Hyp	Ref	Expression
1		elisset 1813	. 2 ⊢ (A ∈ C → A ∈ V)
2		df-clel 1470	. . . . 5 ⊢ ({y∣φ} ∈ B ↔ ∃w(w = {y∣φ} ⋀ w ∈ B))
3	2	sbcbii 1974	. . . 4 ⊢ (A ∈ V → ([A / x]{y∣φ} ∈ B ↔ [A / x]∃w(w = {y∣φ} ⋀ w ∈ B)))
4		sbcexg 1971	. . . 4 ⊢ (A ∈ V → ([A / x]∃w(w = {y∣φ} ⋀ w ∈ B) ↔ ∃w[A / x](w = {y∣φ} ⋀ w ∈ B)))
5		sbcang 1967	. . . . . 6 ⊢ (A ∈ V → ([A / x](w = {y∣φ} ⋀ w ∈ B) ↔ ([A / x]w = {y∣φ} ⋀ [A / x]w ∈ B)))
6		abeq2 1565	. . . . . . . . . 10 ⊢ (w = {y∣φ} ↔ ∀y(y ∈ w ↔ φ))
7	6	sbcbii 1974	. . . . . . . . 9 ⊢ (A ∈ V → ([A / x]w = {y∣φ} ↔ [A / x]∀y(y ∈ w ↔ φ)))
8		sbcalg 1970	. . . . . . . . 9 ⊢ (A ∈ V → ([A / x]∀y(y ∈ w ↔ φ) ↔ ∀y[A / x](y ∈ w ↔ φ)))
9		sbcbidig 1969	. . . . . . . . . . 11 ⊢ (A ∈ V → ([A / x](y ∈ w ↔ φ) ↔ ([A / x]y ∈ w ↔ [A / x]φ)))
10		ax-17 969	. . . . . . . . . . . . 13 ⊢ (y ∈ w → ∀x y ∈ w)
11	10	sbcgf 1982	. . . . . . . . . . . 12 ⊢ (A ∈ V → ([A / x]y ∈ w ↔ y ∈ w))
12	11	bibi1d 618	. . . . . . . . . . 11 ⊢ (A ∈ V → (([A / x]y ∈ w ↔ [A / x]φ) ↔ (y ∈ w ↔ [A / x]φ)))
13	9, 12	bitrd 527	. . . . . . . . . 10 ⊢ (A ∈ V → ([A / x](y ∈ w ↔ φ) ↔ (y ∈ w ↔ [A / x]φ)))
14	13	albidv 1276	. . . . . . . . 9 ⊢ (A ∈ V → (∀y[A / x](y ∈ w ↔ φ) ↔ ∀y(y ∈ w ↔ [A / x]φ)))
15	7, 8, 14	3bitrd 543	. . . . . . . 8 ⊢ (A ∈ V → ([A / x]w = {y∣φ} ↔ ∀y(y ∈ w ↔ [A / x]φ)))
16		abeq2 1565	. . . . . . . 8 ⊢ (w = {y∣[A / x]φ} ↔ ∀y(y ∈ w ↔ [A / x]φ))
17	15, 16	syl6bbr 537	. . . . . . 7 ⊢ (A ∈ V → ([A / x]w = {y∣φ} ↔ w = {y∣[A / x]φ}))
18		ax-17 969	. . . . . . . . 9 ⊢ (z ∈ w → ∀x z ∈ w)
19		sbcabel.1	. . . . . . . . 9 ⊢ (z ∈ B → ∀x z ∈ B)
20	18, 19	hbel 1563	. . . . . . . 8 ⊢ (w ∈ B → ∀x w ∈ B)
21	20	sbcgf 1982	. . . . . . 7 ⊢ (A ∈ V → ([A / x]w ∈ B ↔ w ∈ B))
22	17, 21	anbi12d 627	. . . . . 6 ⊢ (A ∈ V → (([A / x]w = {y∣φ} ⋀ [A / x]w ∈ B) ↔ (w = {y∣[A / x]φ} ⋀ w ∈ B)))
23	5, 22	bitrd 527	. . . . 5 ⊢ (A ∈ V → ([A / x](w = {y∣φ} ⋀ w ∈ B) ↔ (w = {y∣[A / x]φ} ⋀ w ∈ B)))
24	23	exbidv 1277	. . . 4 ⊢ (A ∈ V → (∃w[A / x](w = {y∣φ} ⋀ w ∈ B) ↔ ∃w(w = {y∣[A / x]φ} ⋀ w ∈ B)))
25	3, 4, 24	3bitrd 543	. . 3 ⊢ (A ∈ V → ([A / x]{y∣φ} ∈ B ↔ ∃w(w = {y∣[A / x]φ} ⋀ w ∈ B)))
26		df-clel 1470	. . 3 ⊢ ({y∣[A / x]φ} ∈ B ↔ ∃w(w = {y∣[A / x]φ} ⋀ w ∈ B))
27	25, 26	syl6bbr 537	. 2 ⊢ (A ∈ V → ([A / x]{y∣φ} ∈ B ↔ {y∣[A / x]φ} ∈ B))
28	1, 27	syl 10	1 ⊢ (A ∈ C → ([A / x]{y∣φ} ∈ B ↔ {y∣[A / x]φ} ∈ B))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978  [wsbc 1168  {cab 1461  Vcvv 1807

This theorem is referenced by:  csbexg 2004

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938

Copyright terms: Public domain