Theorem pm13.183 2822

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 2146	. 2 |- ( y = A -> ( y = B <-> A = B ) )
2		eqeq2 2149	. . . 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 1796	. 2 |- ( y = A -> ( A. z ( z = y <-> z = B ) <-> A. z ( z = A <-> z = B ) ) )
5		eqeq2 2149	. . . 4 |- ( y = B -> ( z = y <-> z = B ) )
6	5	alrimiv 1846	. . 3 |- ( y = B -> A. z ( z = y <-> z = B ) )
7		stdpc4 1748	. . . 4 |- ( A. z ( z = y <-> z = B ) -> [ y / z ] ( z = y <-> z = B ) )
8		sbbi 1932	. . . . 5 |- ( [ y / z ] ( z = y <-> z = B ) <-> ( [ y / z ] z = y <-> [ y / z ] z = B ) )
9		eqsb3 2243	. . . . . . 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 1758	. . . . . . 7 |- [ y / z ] z = y
12		bi1 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 2747	1 |- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   e. wcel 1480   [wsb 1735

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  mpo2eqb  5880