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Theorem nfeqd 2503
 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 2347 . 2 (A = By(y Ay B))
2 nfv 1619 . . 3 yφ
3 nfeqd.1 . . . . 5 (φxA)
43nfcrd 2502 . . . 4 (φ → Ⅎx y A)
5 nfeqd.2 . . . . 5 (φxB)
65nfcrd 2502 . . . 4 (φ → Ⅎx y B)
74, 6nfbid 1832 . . 3 (φ → Ⅎx(y Ay B))
82, 7nfald 1852 . 2 (φ → Ⅎxy(y Ay B))
91, 8nfxfrd 1571 1 (φ → Ⅎx A = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-nfc 2478 This theorem is referenced by:  nfeld  2504  nfned  2612  vtoclgft  2905  sbcralt  3118  csbiebt  3172  dfnfc2  3909  nfiotad  4342  iota2df  4365  dfid3  4768  oprabid  5550
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