Theorem pm13.183 3344

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 2626	. 2 |- ( y = A -> ( y = B <-> A = B ) )
2		eqeq2 2633	. . . 4 |- ( y = A -> ( z = y <-> z = A ) )
3	2	bibi1d 333	. . 3 |- ( y = A -> ( ( z = y <-> z = B ) <-> ( z = A <-> z = B ) ) )
4	3	albidv 1849	. 2 |- ( y = A -> ( A. z ( z = y <-> z = B ) <-> A. z ( z = A <-> z = B ) ) )
5		eqeq2 2633	. . . 4 |- ( y = B -> ( z = y <-> z = B ) )
6	5	alrimiv 1855	. . 3 |- ( y = B -> A. z ( z = y <-> z = B ) )
7		stdpc4 2353	. . . 4 |- ( A. z ( z = y <-> z = B ) -> [ y / z ] ( z = y <-> z = B ) )
8		sbbi 2401	. . . . 5 |- ( [ y / z ] ( z = y <-> z = B ) <-> ( [ y / z ] z = y <-> [ y / z ] z = B ) )
9		eqsb3 2728	. . . . . . 7 |- ( [ y / z ] z = B <-> y = B )
10	9	bibi2i 327	. . . . . 6 |- ( ( [ y / z ] z = y <-> [ y / z ] z = B ) <-> ( [ y / z ] z = y <-> y = B ) )
11		equsb1 2368	. . . . . . 7 |- [ y / z ] z = y
12		biimp 205	. . . . . . 7 |- ( ( [ y / z ] z = y <-> y = B ) -> ( [ y / z ] z = y -> y = B ) )
13	11, 12	mpi 20	. . . . . 6 |- ( ( [ y / z ] z = y <-> y = B ) -> y = B )
14	10, 13	sylbi 207	. . . . 5 |- ( ( [ y / z ] z = y <-> [ y / z ] z = B ) -> y = B )
15	8, 14	sylbi 207	. . . 4 |- ( [ y / z ] ( z = y <-> z = B ) -> y = B )
16	7, 15	syl 17	. . 3 |- ( A. z ( z = y <-> z = B ) -> y = B )
17	6, 16	impbii 199	. 2 |- ( y = B <-> A. z ( z = y <-> z = B ) )
18	1, 4, 17	vtoclbg 3267	1 |- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   [wsb 1880   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  mpt22eqb  6769