Theorem sbceqdig 2008

Description: Distribute proper substitution through an equality relation.

Assertion

Ref	Expression
sbceqdig	⊢ (A ∈ D → ([A / x]B = C ↔ [A / x]B = [A / x]C))

Proof of Theorem sbceqdig

Step	Hyp	Ref	Expression
1		elisset 1813	. . 3 ⊢ (A ∈ D → A ∈ V)
2		sbcalg 1970	. . . . 5 ⊢ (A ∈ V → ([A / x]∀z(z ∈ B ↔ z ∈ C) ↔ ∀z[A / x](z ∈ B ↔ z ∈ C)))
3		dfcleq 1468	. . . . . 6 ⊢ (B = C ↔ ∀z(z ∈ B ↔ z ∈ C))
4	3	sbcbii 1974	. . . . 5 ⊢ (A ∈ V → ([A / x]B = C ↔ [A / x]∀z(z ∈ B ↔ z ∈ C)))
5		eleq1 1531	. . . . . . . . . . . 12 ⊢ (y = z → (y ∈ B ↔ z ∈ B))
6	5	sbcbidv 1973	. . . . . . . . . . 11 ⊢ ((y = z ⋀ A ∈ V) → ([A / x]y ∈ B ↔ [A / x]z ∈ B))
7	6	expcom 374	. . . . . . . . . 10 ⊢ (A ∈ V → (y = z → ([A / x]y ∈ B ↔ [A / x]z ∈ B)))
8	7	19.21aiv 1284	. . . . . . . . 9 ⊢ (A ∈ V → ∀y(y = z → ([A / x]y ∈ B ↔ [A / x]z ∈ B)))
9		visset 1809	. . . . . . . . . 10 ⊢ z ∈ V
10		elabgt 1891	. . . . . . . . . 10 ⊢ ((z ∈ V ⋀ ∀y(y = z → ([A / x]y ∈ B ↔ [A / x]z ∈ B))) → (z ∈ {y∣[A / x]y ∈ B} ↔ [A / x]z ∈ B))
11	9, 10	mpan 694	. . . . . . . . 9 ⊢ (∀y(y = z → ([A / x]y ∈ B ↔ [A / x]z ∈ B)) → (z ∈ {y∣[A / x]y ∈ B} ↔ [A / x]z ∈ B))
12	8, 11	syl 10	. . . . . . . 8 ⊢ (A ∈ V → (z ∈ {y∣[A / x]y ∈ B} ↔ [A / x]z ∈ B))
13		eleq1 1531	. . . . . . . . . . . 12 ⊢ (y = z → (y ∈ C ↔ z ∈ C))
14	13	sbcbidv 1973	. . . . . . . . . . 11 ⊢ ((y = z ⋀ A ∈ V) → ([A / x]y ∈ C ↔ [A / x]z ∈ C))
15	14	expcom 374	. . . . . . . . . 10 ⊢ (A ∈ V → (y = z → ([A / x]y ∈ C ↔ [A / x]z ∈ C)))
16	15	19.21aiv 1284	. . . . . . . . 9 ⊢ (A ∈ V → ∀y(y = z → ([A / x]y ∈ C ↔ [A / x]z ∈ C)))
17		elabgt 1891	. . . . . . . . . 10 ⊢ ((z ∈ V ⋀ ∀y(y = z → ([A / x]y ∈ C ↔ [A / x]z ∈ C))) → (z ∈ {y∣[A / x]y ∈ C} ↔ [A / x]z ∈ C))
18	9, 17	mpan 694	. . . . . . . . 9 ⊢ (∀y(y = z → ([A / x]y ∈ C ↔ [A / x]z ∈ C)) → (z ∈ {y∣[A / x]y ∈ C} ↔ [A / x]z ∈ C))
19	16, 18	syl 10	. . . . . . . 8 ⊢ (A ∈ V → (z ∈ {y∣[A / x]y ∈ C} ↔ [A / x]z ∈ C))
20	12, 19	bibi12d 628	. . . . . . 7 ⊢ (A ∈ V → ((z ∈ {y∣[A / x]y ∈ B} ↔ z ∈ {y∣[A / x]y ∈ C}) ↔ ([A / x]z ∈ B ↔ [A / x]z ∈ C)))
21		sbcbidig 1969	. . . . . . 7 ⊢ (A ∈ V → ([A / x](z ∈ B ↔ z ∈ C) ↔ ([A / x]z ∈ B ↔ [A / x]z ∈ C)))
22	20, 21	bitr4d 530	. . . . . 6 ⊢ (A ∈ V → ((z ∈ {y∣[A / x]y ∈ B} ↔ z ∈ {y∣[A / x]y ∈ C}) ↔ [A / x](z ∈ B ↔ z ∈ C)))
23	22	albidv 1276	. . . . 5 ⊢ (A ∈ V → (∀z(z ∈ {y∣[A / x]y ∈ B} ↔ z ∈ {y∣[A / x]y ∈ C}) ↔ ∀z[A / x](z ∈ B ↔ z ∈ C)))
24	2, 4, 23	3bitr4d 549	. . . 4 ⊢ (A ∈ V → ([A / x]B = C ↔ ∀z(z ∈ {y∣[A / x]y ∈ B} ↔ z ∈ {y∣[A / x]y ∈ C})))
25		dfcleq 1468	. . . 4 ⊢ ({y∣[A / x]y ∈ B} = {y∣[A / x]y ∈ C} ↔ ∀z(z ∈ {y∣[A / x]y ∈ B} ↔ z ∈ {y∣[A / x]y ∈ C}))
26	24, 25	syl6bbr 537	. . 3 ⊢ (A ∈ V → ([A / x]B = C ↔ {y∣[A / x]y ∈ B} = {y∣[A / x]y ∈ C}))
27	1, 26	syl 10	. 2 ⊢ (A ∈ D → ([A / x]B = C ↔ {y∣[A / x]y ∈ B} = {y∣[A / x]y ∈ C}))
28		df-csb 1998	. . 3 ⊢ [A / x]B = {y∣[A / x]y ∈ B}
29		df-csb 1998	. . 3 ⊢ [A / x]C = {y∣[A / x]y ∈ C}
30	28, 29	eqeq12i 1485	. 2 ⊢ ([A / x]B = [A / x]C ↔ {y∣[A / x]y ∈ B} = {y∣[A / x]y ∈ C})
31	27, 30	syl6bbr 537	1 ⊢ (A ∈ D → ([A / x]B = C ↔ [A / x]B = [A / x]C))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 952   = wceq 954   ∈ wcel 956  [wsbc 1168  {cab 1461  Vcvv 1807  [csb 1997

This theorem is referenced by:  sbceq1dig 2010  sbceq2dig 2012  csbeq2d 2014  csbeq2i 2016  fsum1s 6955  fsump1s 6959  csbfsum 6973

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-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-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938  df-csb 1998

Copyright terms: Public domain