Theorem sbabel 1189

Description: Theorem to move a substitution in and out of a class abstraction.

Hypothesis

Ref	Expression
sbabel.1	|- (w e. A -> A.x w e. A)

Assertion

Ref	Expression
sbabel	|- ([y / x]{z | ph} e. A <-> {z | [y / x]ph} e. A)

Distinct variable group(s):   w,A   x,w   x,z   y,z

Proof of Theorem sbabel

Step	Hyp	Ref	Expression
1		sbex 998	. . 3 |- ([y / x]E.v(v = {z | ph} /\ v e. A) <-> E.v[y / x](v = {z | ph} /\ v e. A))
2		sban 889	. . . . 5 |- ([y / x](v = {z | ph} /\ v e. A) <-> ([y / x]v = {z | ph} /\ [y / x]v e. A))
3		ax-17 925	. . . . . . . . . 10 |- (w e. v -> A.x w e. v)
4		sbabel.1	. . . . . . . . . 10 |- (w e. A -> A.x w e. A)
5	3, 4	hbel 1172	. . . . . . . . 9 |- (v e. A -> A.x v e. A)
6	5	sbf 870	. . . . . . . 8 |- ([y / x]v e. A <-> v e. A)
7	6	anbi2i 367	. . . . . . 7 |- (([y / x]v = {z | ph} /\ [y / x]v e. A) <-> ([y / x]v = {z | ph} /\ v e. A))
8	7	bicomi 150	. . . . . 6 |- (([y / x]v = {z | ph} /\ v e. A) <-> ([y / x]v = {z | ph} /\ [y / x]v e. A))
9		sbal 997	. . . . . . . . 9 |- ([y / x]A.z(z e. v <-> ph) <-> A.z[y / x](z e. v <-> ph))
10		sbbi 890	. . . . . . . . . . 11 |- ([y / x](z e. v <-> ph) <-> ([y / x]z e. v <-> [y / x]ph))
11		ax-17 925	. . . . . . . . . . . . . 14 |- (z e. v -> A.x z e. v)
12	11	sbf 870	. . . . . . . . . . . . 13 |- ([y / x]z e. v <-> z e. v)
13	12	bibi1i 461	. . . . . . . . . . . 12 |- (([y / x]z e. v <-> [y / x]ph) <-> (z e. v <-> [y / x]ph))
14	13	bicomi 150	. . . . . . . . . . 11 |- ((z e. v <-> [y / x]ph) <-> ([y / x]z e. v <-> [y / x]ph))
15	10, 14	bitr4 154	. . . . . . . . . 10 |- ([y / x](z e. v <-> ph) <-> (z e. v <-> [y / x]ph))
16	15	bial 695	. . . . . . . . 9 |- (A.z[y / x](z e. v <-> ph) <-> A.z(z e. v <-> [y / x]ph))
17	9, 16	bitr 151	. . . . . . . 8 |- ([y / x]A.z(z e. v <-> ph) <-> A.z(z e. v <-> [y / x]ph))
18		cleqab 1174	. . . . . . . . 9 |- (v = {z | ph} <-> A.z(z e. v <-> ph))
19	18	bisb 855	. . . . . . . 8 |- ([y / x]v = {z | ph} <-> [y / x]A.z(z e. v <-> ph))
20		cleqab 1174	. . . . . . . 8 |- (v = {z | [y / x]ph} <-> A.z(z e. v <-> [y / x]ph))
21	17, 19, 20	3bitr4 158	. . . . . . 7 |- ([y / x]v = {z | ph} <-> v = {z | [y / x]ph})
22	21	anbi1i 368	. . . . . 6 |- (([y / x]v = {z | ph} /\ v e. A) <-> (v = {z | [y / x]ph} /\ v e. A))
23	8, 22	bitr3 153	. . . . 5 |- (([y / x]v = {z | ph} /\ [y / x]v e. A) <-> (v = {z | [y / x]ph} /\ v e. A))
24	2, 23	bitr 151	. . . 4 |- ([y / x](v = {z | ph} /\ v e. A) <-> (v = {z | [y / x]ph} /\ v e. A))
25	24	biex 733	. . 3 |- (E.v[y / x](v = {z | ph} /\ v e. A) <-> E.v(v = {z | [y / x]ph} /\ v e. A))
26	1, 25	bitr 151	. 2 |- ([y / x]E.v(v = {z | ph} /\ v e. A) <-> E.v(v = {z | [y / x]ph} /\ v e. A))
27		df-clel 1099	. . 3 |- ({z | ph} e. A <-> E.v(v = {z | ph} /\ v e. A))
28	27	bisb 855	. 2 |- ([y / x]{z | ph} e. A <-> [y / x]E.v(v = {z | ph} /\ v e. A))
29		df-clel 1099	. 2 |- ({z | [y / x]ph} e. A <-> E.v(v = {z | [y / x]ph} /\ v e. A))
30	26, 28, 29	3bitr4 158	1 |- ([y / x]{z | ph} e. A <-> {z | [y / x]ph} e. A)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   e. wel 803  [wsb 852  {cab 1090   = wceq 1091   e. wcel 1092

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099

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