Theorem pm13.183 2675

Description: Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only A is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.183	|- (A e. V -> (A = B <-> A.z(z = A <-> z = B)))

Distinct variable groups:   z,A   z,B

Allowed substitution hint:   V(z)

Proof of Theorem pm13.183

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2043	. 2 |- (y = A -> (y = B <-> A = B))
2		eqeq2 2046	. . . 4 |- (y = A -> (z = y <-> z = A))
3	2	bibi1d 222	. . 3 |- (y = A -> ((z = y <-> z = B) <-> (z = A <-> z = B)))
4	3	albidv 1702	. 2 |- (y = A -> (A.z(z = y <-> z = B) <-> A.z(z = A <-> z = B)))
5		eqeq2 2046	. . . 4 |- (y = B -> (z = y <-> z = B))
6	5	alrimiv 1751	. . 3 |- (y = B -> A.z(z = y <-> z = B))
7		stdpc4 1655	. . . 4 |- (A.z(z = y <-> z = B) -> [y / z](z = y <-> z = B))
8		sbbi 1830	. . . . 5 |- ([y / z](z = y <-> z = B) <-> ([y / z]z = y <-> [y / z]z = B))
9		eqsb3 2138	. . . . . . 7 |- ([y / z]z = B <-> y = B)
10	9	bibi2i 216	. . . . . 6 |- (([y / z]z = y <-> [y / z]z = B) <-> ([y / z]z = y <-> y = B))
11		equsb1 1665	. . . . . . 7 |- [y / z]z = y
12		bi1 111	. . . . . . 7 |- (([y / z]z = y <-> y = B) -> ([y / z]z = y -> y = B))
13	11, 12	mpi 15	. . . . . 6 |- (([y / z]z = y <-> y = B) -> y = B)
14	10, 13	sylbi 114	. . . . 5 |- (([y / z]z = y <-> [y / z]z = B) -> y = B)
15	8, 14	sylbi 114	. . . 4 |- ([y / z](z = y <-> z = B) -> y = B)
16	7, 15	syl 14	. . 3 |- (A.z(z = y <-> z = B) -> y = B)
17	6, 16	impbii 117	. 2 |- (y = B <-> A.z(z = y <-> z = B))
18	1, 4, 17	vtoclbg 2608	1 |- (A e. V -> (A = B <-> A.z(z = A <-> z = B)))

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240   = wceq 1242   e. wcel 1390  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553

This theorem is referenced by:  mpt22eqb  5552