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Theorem nfeq 2185
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 2034 . 2  |-  ( A  =  B  <->  A. z
( z  e.  A  <->  z  e.  B ) )
2 nfnfc.1 . . . . 5  |-  F/_ x A
32nfcri 2172 . . . 4  |-  F/ x  z  e.  A
4 nfeq.2 . . . . 5  |-  F/_ x B
54nfcri 2172 . . . 4  |-  F/ x  z  e.  B
63, 5nfbi 1481 . . 3  |-  F/ x
( z  e.  A  <->  z  e.  B )
76nfal 1468 . 2  |-  F/ x A. z ( z  e.  A  <->  z  e.  B
)
81, 7nfxfr 1363 1  |-  F/ x  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 98   A.wal 1241    = wceq 1243   F/wnf 1349    e. wcel 1393   F/_wnfc 2165
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167
This theorem is referenced by:  nfel  2186  nfeq1  2187  nfeq2  2189  nfne  2297  raleqf  2501  rexeqf  2502  reueq1f  2503  rmoeq1f  2504  rabeqf  2550  sbceqg  2866  csbhypf  2885  nfiotadxy  4870  nffn  4995  nffo  5105  fvmptdf  5258  mpteqb  5261  fvmptf  5263  eqfnfv2f  5269  dff13f  5409  ovmpt2s  5624  ov2gf  5625  ovmpt2dxf  5626  ovmpt2df  5632  eqerlem  6137
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