Theorem hbint 2547

Description: Bound-variable hypothesis builder for intersection.

Hypothesis

Ref	Expression
hbint.1	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbint	|- (y e. |^|A -> A.x y e. |^|A)

Distinct variable groups:   y,A   x,y

Proof of Theorem hbint

Step	Hyp	Ref	Expression
1		ax-17 973	. . . . 5 |- (y e. z -> A.x y e. z)
2		hbint.1	. . . . 5 |- (y e. A -> A.x y e. A)
3	1, 2	hbel 1569	. . . 4 |- (z e. A -> A.x z e. A)
4	3, 1	hbim 1009	. . 3 |- ((z e. A -> y e. z) -> A.x(z e. A -> y e. z))
5	4	hbal 1007	. 2 |- (A.z(z e. A -> y e. z) -> A.xA.z(z e. A -> y e. z))
6		visset 1816	. . 3 |- y e. V
7	6	elint 2543	. 2 |- (y e. |^|A <-> A.z(z e. A -> y e. z))
8	7	albii 1001	. 2 |- (A.x y e. |^|A <-> A.xA.z(z e. A -> y e. z))
9	5, 7, 8	3imtr4 219	1 |- (y e. |^|A -> A.x y e. |^|A)

Syntax hints:   -> wi 3  A.wal 956   e. wcel 960  |^|cint 2537

This theorem is referenced by:  intab 2564  onminsb 3015  onminex 3026  oawordeulem 4194  unblem2 4552  unblem3 4553  tz9.12lem3 4671  rankid 4682  cardmin 4871  alephordlem1 4883  cardaleph 4896

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-v 1815  df-int 2538

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