Theorem elex22 2303

Description: If two classes each contain another class, then both contain some set. This proof was automatically generated from the virtual deduction proof elex22VD 16663 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
elex22	|- ((A e. B /\ A e. C) -> E.x(x e. B /\ x e. C))

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem elex22

Step	Hyp	Ref	Expression
1		eleq1a 1966	. . . 4 |- (A e. B -> (x = A -> x e. B))
2		eleq1a 1966	. . . 4 |- (A e. C -> (x = A -> x e. C))
3	1, 2	anim12ii 618	. . 3 |- ((A e. B /\ A e. C) -> (x = A -> (x e. B /\ x e. C)))
4	3	19.21aiv 1664	. 2 |- ((A e. B /\ A e. C) -> A.x(x = A -> (x e. B /\ x e. C)))
5		elex 2302	. . 3 |- (A e. B -> E.x x = A)
6	5	adantr 425	. 2 |- ((A e. B /\ A e. C) -> E.x x = A)
7		exim 1386	. 2 |- (A.x(x = A -> (x e. B /\ x e. C)) -> (E.x x = A -> E.x(x e. B /\ x e. C)))
8	4, 6, 7	sylc 83	1 |- ((A e. B /\ A e. C) -> E.x(x e. B /\ x e. C))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326

This theorem is referenced by:  en3lplem1 5756  en3lplem1VD 16667

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

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

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