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Theorem nfeq 2289
Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
nfnfc.1  |-  F/_ x A
nfeq.2  |-  F/_ x B
Assertion
Ref Expression
nfeq  |-  F/ x  A  =  B

Proof of Theorem nfeq
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2133 . 2  |-  ( A  =  B  <->  A. z
( z  e.  A  <->  z  e.  B ) )
2 nfnfc.1 . . . . 5  |-  F/_ x A
32nfcri 2275 . . . 4  |-  F/ x  z  e.  A
4 nfeq.2 . . . . 5  |-  F/_ x B
54nfcri 2275 . . . 4  |-  F/ x  z  e.  B
63, 5nfbi 1568 . . 3  |-  F/ x
( z  e.  A  <->  z  e.  B )
76nfal 1555 . 2  |-  F/ x A. z ( z  e.  A  <->  z  e.  B
)
81, 7nfxfr 1450 1  |-  F/ x  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1329    = wceq 1331   F/wnf 1436    e. wcel 1480   F/_wnfc 2268
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270
This theorem is referenced by:  nfel  2290  nfeq1  2291  nfeq2  2293  nfne  2401  raleqf  2622  rexeqf  2623  reueq1f  2624  rmoeq1f  2625  rabeqf  2676  sbceqg  3018  csbhypf  3038  nfiotadw  5091  nffn  5219  nffo  5344  fvmptdf  5508  mpteqb  5511  fvmptf  5513  eqfnfv2f  5522  dff13f  5671  ovmpos  5894  ov2gf  5895  ovmpodxf  5896  ovmpodf  5902  eqerlem  6460  sumeq2  11128  fsumadd  11175  prodeq1f  11321  prodeq2  11326  txcnp  12440  cnmpt11  12452  cnmpt21  12460  cnmptcom  12467
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