Theorem elintrabg 2550

Description: Membership in the intersection of a class abstraction.

Assertion

Ref	Expression
elintrabg	|- (A e. C -> (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x)))

Distinct variable group:   x,A

Proof of Theorem elintrabg

Step	Hyp	Ref	Expression
1		eleq1 1537	. 2 |- (y = A -> (y e. |^|{x e. B | ph} <-> A e. |^|{x e. B | ph}))
2		eleq1 1537	. . . 4 |- (y = A -> (y e. x <-> A e. x))
3	2	imbi2d 614	. . 3 |- (y = A -> ((ph -> y e. x) <-> (ph -> A e. x)))
4	3	ralbidv 1666	. 2 |- (y = A -> (A.x e. B (ph -> y e. x) <-> A.x e. B (ph -> A e. x)))
5		visset 1816	. . 3 |- y e. V
6	5	elintrab 2549	. 2 |- (y e. |^|{x e. B | ph} <-> A.x e. B (ph -> y e. x))
7	1, 4, 6	vtoclbg 1851	1 |- (A e. C -> (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x)))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  A.wral 1648  {crab 1651  |^|cint 2537

This theorem is referenced by:  islp2 7744

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rab 1655  df-v 1815  df-int 2538

Copyright terms: Public domain