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Theorem nfiinya 3810
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunya.1  |-  F/_ y A
nfiunya.2  |-  F/_ y B
Assertion
Ref Expression
nfiinya  |-  F/_ y |^|_ x  e.  A  B
Distinct variable group:    x, A
Allowed substitution hints:    A( y)    B( x, y)

Proof of Theorem nfiinya
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-iin 3784 . 2  |-  |^|_ x  e.  A  B  =  { z  |  A. x  e.  A  z  e.  B }
2 nfiunya.1 . . . 4  |-  F/_ y A
3 nfiunya.2 . . . . 5  |-  F/_ y B
43nfcri 2250 . . . 4  |-  F/ y  z  e.  B
52, 4nfralya 2448 . . 3  |-  F/ y A. x  e.  A  z  e.  B
65nfab 2261 . 2  |-  F/_ y { z  |  A. x  e.  A  z  e.  B }
71, 6nfcxfr 2253 1  |-  F/_ y |^|_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1463   {cab 2101   F/_wnfc 2243   A.wral 2391   |^|_ciin 3782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-iin 3784
This theorem is referenced by: (None)
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