Theorem sbabel 1584

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 groups:   w,A   x,w   x,z   y,z

Proof of Theorem sbabel

Step	Hyp	Ref	Expression
1		sbex 1348	. . 3 |- ([y / x]E.v(v = {z | ph} /\ v e. A) <-> E.v[y / x](v = {z | ph} /\ v e. A))
2		sban 1237	. . . . 5 |- ([y / x](v = {z | ph} /\ v e. A) <-> ([y / x]v = {z | ph} /\ [y / x]v e. A))
3		sbal 1347	. . . . . . . 8 |- ([y / x]A.z(z e. v <-> ph) <-> A.z[y / x](z e. v <-> ph))
4		ax-17 971	. . . . . . . . . . 11 |- (z e. v -> A.x z e. v)
5	4	sbf 1186	. . . . . . . . . 10 |- ([y / x]z e. v <-> z e. v)
6	5	sbrbis 1241	. . . . . . . . 9 |- ([y / x](z e. v <-> ph) <-> (z e. v <-> [y / x]ph))
7	6	albii 999	. . . . . . . 8 |- (A.z[y / x](z e. v <-> ph) <-> A.z(z e. v <-> [y / x]ph))
8	3, 7	bitr 173	. . . . . . 7 |- ([y / x]A.z(z e. v <-> ph) <-> A.z(z e. v <-> [y / x]ph))
9		abeq2 1568	. . . . . . . 8 |- (v = {z | ph} <-> A.z(z e. v <-> ph))
10	9	sbbii 1174	. . . . . . 7 |- ([y / x]v = {z | ph} <-> [y / x]A.z(z e. v <-> ph))
11		abeq2 1568	. . . . . . 7 |- (v = {z | [y / x]ph} <-> A.z(z e. v <-> [y / x]ph))
12	8, 10, 11	3bitr4 183	. . . . . 6 |- ([y / x]v = {z | ph} <-> v = {z | [y / x]ph})
13		ax-17 971	. . . . . . . 8 |- (w e. v -> A.x w e. v)
14		sbabel.1	. . . . . . . 8 |- (w e. A -> A.x w e. A)
15	13, 14	hbel 1566	. . . . . . 7 |- (v e. A -> A.x v e. A)
16	15	sbf 1186	. . . . . 6 |- ([y / x]v e. A <-> v e. A)
17	12, 16	anbi12i 482	. . . . 5 |- (([y / x]v = {z | ph} /\ [y / x]v e. A) <-> (v = {z | [y / x]ph} /\ v e. A))
18	2, 17	bitr 173	. . . 4 |- ([y / x](v = {z | ph} /\ v e. A) <-> (v = {z | [y / x]ph} /\ v e. A))
19	18	exbii 1051	. . 3 |- (E.v[y / x](v = {z | ph} /\ v e. A) <-> E.v(v = {z | [y / x]ph} /\ v e. A))
20	1, 19	bitr 173	. 2 |- ([y / x]E.v(v = {z | ph} /\ v e. A) <-> E.v(v = {z | [y / x]ph} /\ v e. A))
21		df-clel 1472	. . 3 |- ({z | ph} e. A <-> E.v(v = {z | ph} /\ v e. A))
22	21	sbbii 1174	. 2 |- ([y / x]{z | ph} e. A <-> [y / x]E.v(v = {z | ph} /\ v e. A))
23		df-clel 1472	. 2 |- ({z | [y / x]ph} e. A <-> E.v(v = {z | [y / x]ph} /\ v e. A))
24	20, 22, 23	3bitr4 183	1 |- ([y / x]{z | ph} e. A <-> {z | [y / x]ph} e. A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170  {cab 1463

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472

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