Theorem sbceqal 3097

Description: Set theory version of sbeqal1 in set.mm. (Contributed by Andrew Salmon, 28-Jun-2011.)

Assertion

Ref	Expression
sbceqal	|- (A e. V -> (A.x(x = A -> x = B) -> A = B))

Distinct variable groups:   x,B   x,A

Allowed substitution hint:   V(x)

Proof of Theorem sbceqal

Step	Hyp	Ref	Expression
1		spsbc 3058	. 2 |- (A e. V -> (A.x(x = A -> x = B) -> [.A / x ].(x = A -> x = B)))
2		sbcimg 3087	. . 3 |- (A e. V -> ( [.A / x ].(x = A -> x = B) <-> ( [.A / x ].x = A -> [.A / x ].x = B)))
3		eqid 2353	. . . . 5 |- A = A
4		eqsbc3 3085	. . . . 5 |- (A e. V -> ( [.A / x ].x = A <-> A = A))
5	3, 4	mpbiri 224	. . . 4 |- (A e. V -> [.A / x ].x = A)
6		pm5.5 326	. . . 4 |- ( [.A / x ].x = A -> (( [.A / x ].x = A -> [.A / x ].x = B) <-> [.A / x ].x = B))
7	5, 6	syl 15	. . 3 |- (A e. V -> (( [.A / x ].x = A -> [.A / x ].x = B) <-> [.A / x ].x = B))
8		eqsbc3 3085	. . 3 |- (A e. V -> ( [.A / x ].x = B <-> A = B))
9	2, 7, 8	3bitrd 270	. 2 |- (A e. V -> ( [.A / x ].(x = A -> x = B) <-> A = B))
10	1, 9	sylibd 205	1 |- (A e. V -> (A.x(x = A -> x = B) -> A = B))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710   [.wsbc 3046

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 2478  df-v 2861  df-sbc 3047

This theorem is referenced by:  sbeqalb  3098