Theorem pm13.183 2681

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 2046	. 2 |- ( y = A -> ( y = B <-> A = B ) )
2		eqeq2 2049	. . . 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 1705	. 2 |- ( y = A -> ( A. z ( z = y <-> z = B ) <-> A. z ( z = A <-> z = B ) ) )
5		eqeq2 2049	. . . 4 |- ( y = B -> ( z = y <-> z = B ) )
6	5	alrimiv 1754	. . 3 |- ( y = B -> A. z ( z = y <-> z = B ) )
7		stdpc4 1658	. . . 4 |- ( A. z ( z = y <-> z = B ) -> [ y / z ] ( z = y <-> z = B ) )
8		sbbi 1833	. . . . 5 |- ( [ y / z ] ( z = y <-> z = B ) <-> ( [ y / z ] z = y <-> [ y / z ] z = B ) )
9		eqsb3 2141	. . . . . . 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 1668	. . . . . . 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 2614	1 |- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   [wsb 1645

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  mpt22eqb  5610