Theorem sbeqalb 3055

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 240	. . . . 5 |- ( ( ph <-> x = A ) -> ( ( ph <-> x = B ) <-> ( x = A <-> x = B ) ) )
2	1	biimpa 296	. . . 4 |- ( ( ( ph <-> x = A ) /\ ( ph <-> x = B ) ) -> ( x = A <-> x = B ) )
3	2	biimpd 144	. . 3 |- ( ( ( ph <-> x = A ) /\ ( ph <-> x = B ) ) -> ( x = A -> x = B ) )
4	3	alanimi 1482	. 2 |- ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A. x ( x = A -> x = B ) )
5		sbceqal 3054	. 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 104   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2176

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:  iotaval  5243