Theorem sbceqal 3020

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 2976	. 2 |- ( A e. V -> ( A. x ( x = A -> x = B ) -> [. A / x ]. ( x = A -> x = B ) ) )
2		sbcimg 3006	. . 3 |- ( A e. V -> ( [. A / x ]. ( x = A -> x = B ) <-> ( [. A / x ]. x = A -> [. A / x ]. x = B ) ) )
3		eqid 2177	. . . . 5 |- A = A
4		eqsbc1 3004	. . . . 5 |- ( A e. V -> ( [. A / x ]. x = A <-> A = A ) )
5	3, 4	mpbiri 168	. . . 4 |- ( A e. V -> [. A / x ]. x = A )
6		pm5.5 242	. . . 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 3004	. . 3 |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
9	2, 7, 8	3bitrd 214	. 2 |- ( A e. V -> ( [. A / x ]. ( x = A -> x = B ) <-> A = B ) )
10	1, 9	sylibd 149	1 |- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2148   [.wsbc 2964

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965

This theorem is referenced by:  sbeqalb  3021  snsssn  3763