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Theorem nfeqd 2175
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 2017 . 2 (A = By(y Ay B))
2 nfv 1403 . . 3 yφ
3 nfeqd.1 . . . . 5 (φxA)
43nfcrd 2174 . . . 4 (φ → Ⅎx y A)
5 nfeqd.2 . . . . 5 (φxB)
65nfcrd 2174 . . . 4 (φ → Ⅎx y B)
74, 6nfbid 1464 . . 3 (φ → Ⅎx(y Ay B))
82, 7nfald 1626 . 2 (φ → Ⅎxy(y Ay B))
91, 8nfxfrd 1344 1 (φ → Ⅎx A = B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1226   = wceq 1228  wnf 1329   wcel 1375  wnfc 2148
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 1316  ax-7 1317  ax-gen 1318  ax-4 1382  ax-17 1401  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-cleq 2016  df-nfc 2150
This theorem is referenced by:  nfeld  2176  nfned  2275  vtoclgft  2580  sbcralt  2810  sbcrext  2811  csbiebt  2862  dfnfc2  3571  eusvnfb  4134  eusv2i  4135  iota2df  4816  riota5f  5414
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