Theorem hbint 2538

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-17 969	. . . . 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 1563	. . . 4 |- (z e. A -> A.x z e. A)
4	3, 1	hbim 1005	. . 3 |- ((z e. A -> y e. z) -> A.x(z e. A -> y e. z))
5	4	hbal 1003	. 2 |- (A.z(z e. A -> y e. z) -> A.xA.z(z e. A -> y e. z))
6		visset 1809	. . 3 |- y e. V
7	6	elint 2534	. 2 |- (y e. |^|A <-> A.z(z e. A -> y e. z))
8	7	albii 997	. 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 952   e. wcel 956  |^|cint 2528

This theorem is referenced by:  intab 2555  onminsb 3004  onminex 3015  oawordeulem 4178  unblem2 4524  unblem3 4525  tz9.12lem3 4641  rankid 4652  cardmin 4840  alephordlem1 4852  cardaleph 4865

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-int 2529

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