Theorem sbceqal 3054

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 3010	. 2 |- ( A e. V -> ( A. x ( x = A -> x = B ) -> [. A / x ]. ( x = A -> x = B ) ) )
2		sbcimg 3040	. . 3 |- ( A e. V -> ( [. A / x ]. ( x = A -> x = B ) <-> ( [. A / x ]. x = A -> [. A / x ]. x = B ) ) )
3		eqid 2205	. . . . 5 |- A = A
4		eqsbc1 3038	. . . . 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 3038	. . 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 1371   = wceq 1373   e. wcel 2176   [.wsbc 2998

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999

This theorem is referenced by:  sbeqalb  3055  snsssn  3802