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Theorem nfeqd 2334
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 2171 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1528 . . 3 𝑦𝜑
3 nfeqd.1 . . . . 5 (𝜑𝑥𝐴)
43nfcrd 2333 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
5 nfeqd.2 . . . . 5 (𝜑𝑥𝐵)
65nfcrd 2333 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐵)
74, 6nfbid 1588 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝑦𝐵))
82, 7nfald 1760 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝑦𝐵))
91, 8nfxfrd 1475 1 (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351   = wceq 1353  wnf 1460  wcel 2148  wnfc 2306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-nfc 2308
This theorem is referenced by:  nfeld  2335  nfned  2441  vtoclgft  2787  sbcralt  3039  sbcrext  3040  csbiebt  3096  dfnfc2  3825  eusvnfb  4451  eusv2i  4452  iota2df  5198  riota5f  5849
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