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Theorem nfeqd 2168
Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfeqd.1 (φxA)
nfeqd.2 (φxB)
Assertion
Ref Expression
nfeqd (φ → Ⅎx A = B)

Proof of Theorem nfeqd
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2010 . 2 (A = By(y Ay B))
2 nfv 1397 . . 3 yφ
3 nfeqd.1 . . . . 5 (φxA)
43nfcrd 2167 . . . 4 (φ → Ⅎx y A)
5 nfeqd.2 . . . . 5 (φxB)
65nfcrd 2167 . . . 4 (φ → Ⅎx y B)
74, 6nfbid 1456 . . 3 (φ → Ⅎx(y Ay B))
82, 7nfald 1619 . 2 (φ → Ⅎxy(y Ay B))
91, 8nfxfrd 1340 1 (φ → Ⅎx A = B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1224   = wceq 1226  wnf 1325   wcel 1369  wnfc 2141
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-5 1312  ax-7 1313  ax-gen 1314  ax-4 1376  ax-17 1395  ax-ial 1403  ax-i5r 1404  ax-ext 1998
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-cleq 2009  df-nfc 2143
This theorem is referenced by:  nfeld  2169  nfned  2270  vtoclgft  2575  sbcralt  2805  sbcrext  2806  csbiebt  2857  dfnfc2  3564  eusvnfb  4127  eusv2i  4128  iota2df  4809  riota5f  5407
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