Theorem elintgOLD 4484

Description: Obsolete proof of elintg 4483 as of 26-Jul-2021. (Contributed by NM, 20-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elintgOLD	|- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem elintgOLD

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. 2 |- ( y = A -> ( y e. |^| B <-> A e. |^| B ) )
2		eleq1 2689	. . 3 |- ( y = A -> ( y e. x <-> A e. x ) )
3	2	ralbidv 2986	. 2 |- ( y = A -> ( A. x e. B y e. x <-> A. x e. B A e. x ) )
4		vex 3203	. . 3 |- y e. _V
5	4	elint2 4482	. 2 |- ( y e. |^| B <-> A. x e. B y e. x )
6	1, 3, 5	vtoclbg 3267	1 |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-int 4476

This theorem is referenced by: (None)