Theorem sbeqalb 2993

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 1439	. 2 |- ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A. x ( x = A -> x = B ) )
5		sbceqal 2992	. 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 1333   = wceq 1335   e. wcel 2128

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938

This theorem is referenced by:  iotaval  5148