Theorem elex22VD 16663

Description: Virtual deduction proof of elex22 2303.

Assertion

Ref	Expression
elex22VD	|- ((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 elex22VD

Step	Hyp	Ref	Expression
1		idn1 16484	. . . . 5 |- . (A e. B /\ A e. C)   ⊢   (A e. B /\ A e. C) .
2		simpl 346	. . . . 5 |- ((A e. B /\ A e. C) -> A e. B)
3	1, 2	e1_ 16518	. . . 4 |- . (A e. B /\ A e. C)   ⊢   A e. B .
4		elex 2302	. . . 4 |- (A e. B -> E.x x = A)
5	3, 4	e1_ 16518	. . 3 |- . (A e. B /\ A e. C)   ⊢   E.x x = A .
6		idn2 16509	. . . . . . . 8 |- . (A e. B /\ A e. C), x = A   ⊢   x = A .
7		eleq1a 1966	. . . . . . . 8 |- (A e. B -> (x = A -> x e. B))
8	3, 6, 7	e12 16593	. . . . . . 7 |- . (A e. B /\ A e. C), x = A   ⊢   x e. B .
9		simpr 350	. . . . . . . . 9 |- ((A e. B /\ A e. C) -> A e. C)
10	1, 9	e1_ 16518	. . . . . . . 8 |- . (A e. B /\ A e. C)   ⊢   A e. C .
11		eleq1a 1966	. . . . . . . 8 |- (A e. C -> (x = A -> x e. C))
12	10, 6, 11	e12 16593	. . . . . . 7 |- . (A e. B /\ A e. C), x = A   ⊢   x e. C .
13		pm3.2 305	. . . . . . 7 |- (x e. B -> (x e. C -> (x e. B /\ x e. C)))
14	8, 12, 13	e22 16561	. . . . . 6 |- . (A e. B /\ A e. C), x = A   ⊢   (x e. B /\ x e. C) .
15	14	in2 16506	. . . . 5 |- . (A e. B /\ A e. C)   ⊢   (x = A -> (x e. B /\ x e. C)) .
16	15	gen11 16511	. . . 4 |- . (A e. B /\ A e. C)   ⊢   A.x(x = A -> (x e. B /\ x e. C)) .
17		exim 1386	. . . 4 |- (A.x(x = A -> (x e. B /\ x e. C)) -> (E.x x = A -> E.x(x e. B /\ x e. C)))
18	16, 17	e1_ 16518	. . 3 |- . (A e. B /\ A e. C)   ⊢   (E.x x = A -> E.x(x e. B /\ x e. C)) .
19		pm2.27 76	. . 3 |- (E.x x = A -> ((E.x x = A -> E.x(x e. B /\ x e. C)) -> E.x(x e. B /\ x e. C)))
20	5, 18, 19	e11 16578	. 2 |- . (A e. B /\ A e. C)   ⊢   E.x(x e. B /\ x e. C) .
21	20	in1 16481	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 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  df-vd1 16480  df-vd2 16489

Copyright terms: Public domain