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Theorem nfeqd 2174
 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 2016 . 2 (A = By(y Ay B))
2 nfv 1402 . . 3 yφ
3 nfeqd.1 . . . . 5 (φxA)
43nfcrd 2173 . . . 4 (φ → Ⅎx y A)
5 nfeqd.2 . . . . 5 (φxB)
65nfcrd 2173 . . . 4 (φ → Ⅎx y B)
74, 6nfbid 1462 . . 3 (φ → Ⅎx(y Ay B))
82, 7nfald 1625 . 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  Ⅎwnf 1329   ∈ wcel 1374  Ⅎwnfc 2147 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 1381  ax-17 1400  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-cleq 2015  df-nfc 2149 This theorem is referenced by:  nfeld  2175  nfned  2274  vtoclgft  2579  sbcralt  2809  sbcrext  2810  csbiebt  2861  dfnfc2  3570  eusvnfb  4134  eusv2i  4135  iota2df  4816  riota5f  5414
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