Theorem sbceqal 3010

Description: A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)

Assertion

Ref	Expression
sbceqal	|- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) )

Distinct variable groups:   x, B   x, A

Allowed substitution hint:   V( x)

Proof of Theorem sbceqal

Step	Hyp	Ref	Expression
1		spsbc 2966	. 2 |- ( A e. V -> ( A. x ( x = A -> x = B ) -> [. A / x ]. ( x = A -> x = B ) ) )
2		sbcimg 2996	. . 3 |- ( A e. V -> ( [. A / x ]. ( x = A -> x = B ) <-> ( [. A / x ]. x = A -> [. A / x ]. x = B ) ) )
3		eqid 2170	. . . . 5 |- A = A
4		eqsbc1 2994	. . . . 5 |- ( A e. V -> ( [. A / x ]. x = A <-> A = A ) )
5	3, 4	mpbiri 167	. . . 4 |- ( A e. V -> [. A / x ]. x = A )
6		pm5.5 241	. . . 4 |- ( [. A / x ]. x = A -> ( ( [. A / x ]. x = A -> [. A / x ]. x = B ) <-> [. A / x ]. x = B ) )
7	5, 6	syl 14	. . 3 |- ( A e. V -> ( ( [. A / x ]. x = A -> [. A / x ]. x = B ) <-> [. A / x ]. x = B ) )
8		eqsbc1 2994	. . 3 |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
9	2, 7, 8	3bitrd 213	. 2 |- ( A e. V -> ( [. A / x ]. ( x = A -> x = B ) <-> A = B ) )
10	1, 9	sylibd 148	1 |- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   [.wsbc 2955

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956

This theorem is referenced by:  sbeqalb  3011  snsssn  3748