Theorem pm13.183

Description: Compare theorem *13.183 in WhiteheadRussell p. 178. Only A is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011) Avoid ax-13 . (Revised by Wolf Lammen, 29-Apr-2023)

	Ref	Expression
Assertion	pm13.183	|- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq1	|- ( y = A -> ( y = B <-> A = B ) )
2		eqeq2	|- ( y = A -> ( z = y <-> z = A ) )
3	2	bibi1d	|- ( y = A -> ( ( z = y <-> z = B ) <-> ( z = A <-> z = B ) ) )
4	3	albidv	|- ( y = A -> ( A. z ( z = y <-> z = B ) <-> A. z ( z = A <-> z = B ) ) )
5		eqeq2	|- ( y = B -> ( z = y <-> z = B ) )
6	5	alrimiv	|- ( y = B -> A. z ( z = y <-> z = B ) )
7		stdpc4	|- ( A. z ( z = y <-> z = B ) -> [ y / z ] ( z = y <-> z = B ) )
8		sbbi	|- ( [ y / z ] ( z = y <-> z = B ) <-> ( [ y / z ] z = y <-> [ y / z ] z = B ) )
9		eqsb1	|- ( [ y / z ] z = B <-> y = B )
10	9	bibi2i	|- ( ( [ y / z ] z = y <-> [ y / z ] z = B ) <-> ( [ y / z ] z = y <-> y = B ) )
11		equsb1v	|- [ y / z ] z = y
12		biimp	|- ( ( [ y / z ] z = y <-> y = B ) -> ( [ y / z ] z = y -> y = B ) )
13	11 12	mpi	|- ( ( [ y / z ] z = y <-> y = B ) -> y = B )
14	10 13	sylbi	|- ( ( [ y / z ] z = y <-> [ y / z ] z = B ) -> y = B )
15	8 14	sylbi	|- ( [ y / z ] ( z = y <-> z = B ) -> y = B )
16	7 15	syl	|- ( A. z ( z = y <-> z = B ) -> y = B )
17	6 16	impbii	|- ( y = B <-> A. z ( z = y <-> z = B ) )
18	1 4 17	vtoclbg	|- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) )