Theorem sbceqal 3030

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 2986	. 2 |- ( A e. V -> ( A. x ( x = A -> x = B ) -> [. A / x ]. ( x = A -> x = B ) ) )
2		sbcimg 3016	. . 3 |- ( A e. V -> ( [. A / x ]. ( x = A -> x = B ) <-> ( [. A / x ]. x = A -> [. A / x ]. x = B ) ) )
3		eqid 2187	. . . . 5 |- A = A
4		eqsbc1 3014	. . . . 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 3014	. . 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 1361   = wceq 1363   e. wcel 2158   [.wsbc 2974

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975

This theorem is referenced by:  sbeqalb  3031  snsssn  3773