Theorem pm13.183 2864

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 2172	. 2 |- ( y = A -> ( y = B <-> A = B ) )
2		eqeq2 2175	. . . 4 |- ( y = A -> ( z = y <-> z = A ) )
3	2	bibi1d 232	. . 3 |- ( y = A -> ( ( z = y <-> z = B ) <-> ( z = A <-> z = B ) ) )
4	3	albidv 1812	. 2 |- ( y = A -> ( A. z ( z = y <-> z = B ) <-> A. z ( z = A <-> z = B ) ) )
5		eqeq2 2175	. . . 4 |- ( y = B -> ( z = y <-> z = B ) )
6	5	alrimiv 1862	. . 3 |- ( y = B -> A. z ( z = y <-> z = B ) )
7		stdpc4 1763	. . . 4 |- ( A. z ( z = y <-> z = B ) -> [ y / z ] ( z = y <-> z = B ) )
8		sbbi 1947	. . . . 5 |- ( [ y / z ] ( z = y <-> z = B ) <-> ( [ y / z ] z = y <-> [ y / z ] z = B ) )
9		eqsb1 2270	. . . . . . 7 |- ( [ y / z ] z = B <-> y = B )
10	9	bibi2i 226	. . . . . 6 |- ( ( [ y / z ] z = y <-> [ y / z ] z = B ) <-> ( [ y / z ] z = y <-> y = B ) )
11		equsb1 1773	. . . . . . 7 |- [ y / z ] z = y
12		biimp 117	. . . . . . 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 120	. . . . 5 |- ( ( [ y / z ] z = y <-> [ y / z ] z = B ) -> y = B )
15	8, 14	sylbi 120	. . . 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 125	. 2 |- ( y = B <-> A. z ( z = y <-> z = B ) )
18	1, 4, 17	vtoclbg 2787	1 |- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   [wsb 1750   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  mpo2eqb  5951