Theorem sbeqalb 3099

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 317	. . . . 5 |- ((ph <-> x = A) -> ((ph <-> x = B) <-> (x = A <-> x = B)))
2	1	biimpa 470	. . . 4 |- (((ph <-> x = A) /\ (ph <-> x = B)) -> (x = A <-> x = B))
3	2	biimpd 198	. . 3 |- (((ph <-> x = A) /\ (ph <-> x = B)) -> (x = A -> x = B))
4	3	alanimi 1562	. 2 |- ((A.x(ph <-> x = A) /\ A.x(ph <-> x = B)) -> A.x(x = A -> x = B))
5		sbceqal 3098	. 2 |- (A e. V -> (A.x(x = A -> x = B) -> A = B))
6	4, 5	syl5 28	1 |- (A e. V -> ((A.x(ph <-> x = A) /\ A.x(ph <-> x = B)) -> A = B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  iotaval  4351