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Theorem nfiin 4122
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 4098 . 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 2568 . . . 4  |-  F/ y  z  e.  B
52, 4nfral 2761 . . 3  |-  F/ y A. x  e.  A  z  e.  B
65nfab 2578 . 2  |-  F/_ y { z  |  A. x  e.  A  z  e.  B }
71, 6nfcxfr 2571 1  |-  F/_ y |^|_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1726   {cab 2424   F/_wnfc 2561   A.wral 2707   |^|_ciin 4096
This theorem is referenced by:  iinab  4154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-iin 4098
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