Theorem sbeqalb 2503

Description: Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.)

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem sbeqalb

Step	Hyp	Ref	Expression
1		sbceqal 2502	. 2 |- (A e. C -> (A.x(x = A -> x = B) -> A = B))
2		bibi1 687	. . . . 5 |- ((ph <-> x = A) -> ((ph <-> x = B) <-> (x = A <-> x = B)))
3	2	biimpa 460	. . . 4 |- (((ph <-> x = A) /\ (ph <-> x = B)) -> (x = A <-> x = B))
4		bi1 165	. . . 4 |- ((x = A <-> x = B) -> (x = A -> x = B))
5	3, 4	syl 12	. . 3 |- (((ph <-> x = A) /\ (ph <-> x = B)) -> (x = A -> x = B))
6	5	alanimi 1342	. 2 |- ((A.x(ph <-> x = A) /\ A.x(ph <-> x = B)) -> A.x(x = A -> x = B))
7	1, 6	syl5 20	1 |- (A e. C -> ((A.x(ph <-> x = A) /\ A.x(ph <-> x = B)) -> A = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300

This theorem is referenced by:  iotaval 5096

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

Copyright terms: Public domain