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Theorem tz6.12f 5145
Description: Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
Hypothesis
Ref Expression
tz6.12f.1  F/_ F
Assertion
Ref Expression
tz6.12f 
<. ,  >.  F 
<. ,  >.  F  F `
Distinct variable group:   ,
Allowed substitution hint:    F()

Proof of Theorem tz6.12f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 opeq2 3541 . . . . 5  <. ,  >.  <. ,  >.
21eleq1d 2103 . . . 4  <. ,  >.  F  <. , 
>.  F
3 tz6.12f.1 . . . . . . 7  F/_ F
43nfel2 2187 . . . . . 6  F/
<. ,  >.  F
5 nfv 1418 . . . . . 6  F/
<. ,  >.  F
64, 5, 2cbveu 1921 . . . . 5  <. , 
>.  F 
<. ,  >.  F
76a1i 9 . . . 4  <. , 
>.  F 
<. ,  >.  F
82, 7anbi12d 442 . . 3  <. , 
>.  F  <. , 
>.  F  <. ,  >.  F  <. ,  >.  F
9 eqeq2 2046 . . 3  F `  F `
108, 9imbi12d 223 . 2  <. , 
>.  F  <. , 
>.  F  F ` 
<. ,  >.  F 
<. ,  >.  F  F `
11 tz6.12 5144 . 2 
<. ,  >.  F 
<. ,  >.  F  F `
1210, 11chvarv 1809 1 
<. ,  >.  F 
<. ,  >.  F  F `
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  weu 1897   F/_wnfc 2162   <.cop 3370   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853
This theorem is referenced by: (None)
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