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Theorem nfiin 4088
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiun.1  |-  F/_ y A
nfiun.2  |-  F/_ y B
Assertion
Ref Expression
nfiin  |-  F/_ y |^|_ x  e.  A  B

Proof of Theorem nfiin
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-iin 4064 . 2  |-  |^|_ x  e.  A  B  =  { z  |  A. x  e.  A  z  e.  B }
2 nfiun.1 . . . 4  |-  F/_ y A
3 nfiun.2 . . . . 5  |-  F/_ y B
43nfcri 2542 . . . 4  |-  F/ y  z  e.  B
52, 4nfral 2727 . . 3  |-  F/ y A. x  e.  A  z  e.  B
65nfab 2552 . 2  |-  F/_ y { z  |  A. x  e.  A  z  e.  B }
71, 6nfcxfr 2545 1  |-  F/_ y |^|_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   {cab 2398   F/_wnfc 2535   A.wral 2674   |^|_ciin 4062
This theorem is referenced by:  iinab  4120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-iin 4064
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