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Theorem nfeqd 2189
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 2031 . 2 (A = By(y Ay B))
2 nfv 1418 . . 3 yφ
3 nfeqd.1 . . . . 5 (φxA)
43nfcrd 2188 . . . 4 (φ → Ⅎx y A)
5 nfeqd.2 . . . . 5 (φxB)
65nfcrd 2188 . . . 4 (φ → Ⅎx y B)
74, 6nfbid 1477 . . 3 (φ → Ⅎx(y Ay B))
82, 7nfald 1640 . 2 (φ → Ⅎxy(y Ay B))
91, 8nfxfrd 1361 1 (φ → Ⅎx A = B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242  wnf 1346   wcel 1390  wnfc 2162
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 1333  ax-7 1334  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-nfc 2164
This theorem is referenced by:  nfeld  2190  nfned  2292  vtoclgft  2598  sbcralt  2828  sbcrext  2829  csbiebt  2880  dfnfc2  3589  eusvnfb  4152  eusv2i  4153  iota2df  4834  riota5f  5435
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