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Theorem nfeq 2201
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 2050 . 2 (𝐴 = 𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2 nfnfc.1 . . . . 5 𝑥𝐴
32nfcri 2188 . . . 4 𝑥 𝑧𝐴
4 nfeq.2 . . . . 5 𝑥𝐵
54nfcri 2188 . . . 4 𝑥 𝑧𝐵
63, 5nfbi 1497 . . 3 𝑥(𝑧𝐴𝑧𝐵)
76nfal 1484 . 2 𝑥𝑧(𝑧𝐴𝑧𝐵)
81, 7nfxfr 1379 1 𝑥 𝐴 = 𝐵
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257   = wceq 1259  wnf 1365  wcel 1409  wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183
This theorem is referenced by:  nfel  2202  nfeq1  2203  nfeq2  2205  nfne  2312  raleqf  2518  rexeqf  2519  reueq1f  2520  rmoeq1f  2521  rabeqf  2567  sbceqg  2893  csbhypf  2912  nfiotadxy  4897  nffn  5022  nffo  5132  fvmptdf  5285  mpteqb  5288  fvmptf  5290  eqfnfv2f  5296  dff13f  5436  ovmpt2s  5651  ov2gf  5652  ovmpt2dxf  5653  ovmpt2df  5659  eqerlem  6167
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