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Theorem nfeqd 2243
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 2082 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1466 . . 3 𝑦𝜑
3 nfeqd.1 . . . . 5 (𝜑𝑥𝐴)
43nfcrd 2242 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
5 nfeqd.2 . . . . 5 (𝜑𝑥𝐵)
65nfcrd 2242 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐵)
74, 6nfbid 1525 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝑦𝐵))
82, 7nfald 1690 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝑦𝐵))
91, 8nfxfrd 1409 1 (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wnf 1394  wcel 1438  wnfc 2215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-nfc 2217
This theorem is referenced by:  nfeld  2244  nfned  2349  vtoclgft  2669  sbcralt  2913  sbcrext  2914  csbiebt  2965  dfnfc2  3666  eusvnfb  4267  eusv2i  4268  iota2df  4991  riota5f  5614
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