Theorem sbeqalb 3011

Description: Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)

Assertion

Ref	Expression
sbeqalb	|- ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A = B ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem sbeqalb

Step	Hyp	Ref	Expression
1		bibi1 239	. . . . 5 |- ( ( ph <-> x = A ) -> ( ( ph <-> x = B ) <-> ( x = A <-> x = B ) ) )
2	1	biimpa 294	. . . 4 |- ( ( ( ph <-> x = A ) /\ ( ph <-> x = B ) ) -> ( x = A <-> x = B ) )
3	2	biimpd 143	. . 3 |- ( ( ( ph <-> x = A ) /\ ( ph <-> x = B ) ) -> ( x = A -> x = B ) )
4	3	alanimi 1452	. 2 |- ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A. x ( x = A -> x = B ) )
5		sbceqal 3010	. 2 |- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) )
6	4, 5	syl5 32	1 |- ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141

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:  iotaval  5171