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Theorem nfeq 2290
Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
nfnfc.1 𝑥𝐴
nfeq.2 𝑥𝐵
Assertion
Ref Expression
nfeq 𝑥 𝐴 = 𝐵

Proof of Theorem nfeq
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2134 . 2 (𝐴 = 𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2 nfnfc.1 . . . . 5 𝑥𝐴
32nfcri 2276 . . . 4 𝑥 𝑧𝐴
4 nfeq.2 . . . . 5 𝑥𝐵
54nfcri 2276 . . . 4 𝑥 𝑧𝐵
63, 5nfbi 1569 . . 3 𝑥(𝑧𝐴𝑧𝐵)
76nfal 1556 . 2 𝑥𝑧(𝑧𝐴𝑧𝐵)
81, 7nfxfr 1451 1 𝑥 𝐴 = 𝐵
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1330   = wceq 1332  wnf 1437  wcel 1481  wnfc 2269
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271
This theorem is referenced by:  nfel  2291  nfeq1  2292  nfeq2  2294  nfne  2402  raleqf  2625  rexeqf  2626  reueq1f  2627  rmoeq1f  2628  rabeqf  2679  sbceqg  3022  csbhypf  3042  nfiotadw  5098  nffn  5226  nffo  5351  fvmptdf  5515  mpteqb  5518  fvmptf  5520  eqfnfv2f  5529  dff13f  5678  ovmpos  5901  ov2gf  5902  ovmpodxf  5903  ovmpodf  5909  eqerlem  6467  sumeq2  11159  fsumadd  11206  prodeq1f  11352  prodeq2  11357  txcnp  12477  cnmpt11  12489  cnmpt21  12497  cnmptcom  12504
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