Theorem pm13.183 2941

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 2236	. 2 |- ( y = A -> ( y = B <-> A = B ) )
2		eqeq2 2239	. . . 4 |- ( y = A -> ( z = y <-> z = A ) )
3	2	bibi1d 233	. . 3 |- ( y = A -> ( ( z = y <-> z = B ) <-> ( z = A <-> z = B ) ) )
4	3	albidv 1870	. 2 |- ( y = A -> ( A. z ( z = y <-> z = B ) <-> A. z ( z = A <-> z = B ) ) )
5		eqeq2 2239	. . . 4 |- ( y = B -> ( z = y <-> z = B ) )
6	5	alrimiv 1920	. . 3 |- ( y = B -> A. z ( z = y <-> z = B ) )
7		stdpc4 1821	. . . 4 |- ( A. z ( z = y <-> z = B ) -> [ y / z ] ( z = y <-> z = B ) )
8		sbbi 2010	. . . . 5 |- ( [ y / z ] ( z = y <-> z = B ) <-> ( [ y / z ] z = y <-> [ y / z ] z = B ) )
9		eqsb1 2333	. . . . . . 7 |- ( [ y / z ] z = B <-> y = B )
10	9	bibi2i 227	. . . . . 6 |- ( ( [ y / z ] z = y <-> [ y / z ] z = B ) <-> ( [ y / z ] z = y <-> y = B ) )
11		equsb1 1831	. . . . . . 7 |- [ y / z ] z = y
12		biimp 118	. . . . . . 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 121	. . . . 5 |- ( ( [ y / z ] z = y <-> [ y / z ] z = B ) -> y = B )
15	8, 14	sylbi 121	. . . 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 126	. 2 |- ( y = B <-> A. z ( z = y <-> z = B ) )
18	1, 4, 17	vtoclbg 2862	1 |- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   [wsb 1808   e. wcel 2200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  mpo2eqb  6113