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Theorem nfeqd 2297
 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 2134 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1509 . . 3 𝑦𝜑
3 nfeqd.1 . . . . 5 (𝜑𝑥𝐴)
43nfcrd 2296 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
5 nfeqd.2 . . . . 5 (𝜑𝑥𝐵)
65nfcrd 2296 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐵)
74, 6nfbid 1568 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝑦𝐵))
82, 7nfald 1734 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝑦𝐵))
91, 8nfxfrd 1452 1 (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330   = wceq 1332  Ⅎwnf 1437   ∈ wcel 1481  Ⅎwnfc 2269 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-nfc 2271 This theorem is referenced by:  nfeld  2298  nfned  2403  vtoclgft  2739  sbcralt  2988  sbcrext  2989  csbiebt  3042  dfnfc2  3760  eusvnfb  4381  eusv2i  4382  iota2df  5118  riota5f  5760
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