Theorem hbint 2543

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-17 971	. . . . 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 1566	. . . 4 |- (z e. A -> A.x z e. A)
4	3, 1	hbim 1007	. . 3 |- ((z e. A -> y e. z) -> A.x(z e. A -> y e. z))
5	4	hbal 1005	. 2 |- (A.z(z e. A -> y e. z) -> A.xA.z(z e. A -> y e. z))
6		visset 1813	. . 3 |- y e. V
7	6	elint 2539	. 2 |- (y e. |^|A <-> A.z(z e. A -> y e. z))
8	7	albii 999	. 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 954   e. wcel 958  |^|cint 2533

This theorem is referenced by:  intab 2560  onminsb 3009  onminex 3020  oawordeulem 4188  unblem2 4541  unblem3 4542  tz9.12lem3 4661  rankid 4672  cardmin 4860  alephordlem1 4872  cardaleph 4885

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-int 2534

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