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Theorem nfeqd 2362
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 2198 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1550 . . 3 𝑦𝜑
3 nfeqd.1 . . . . 5 (𝜑𝑥𝐴)
43nfcrd 2361 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
5 nfeqd.2 . . . . 5 (𝜑𝑥𝐵)
65nfcrd 2361 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐵)
74, 6nfbid 1610 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝑦𝐵))
82, 7nfald 1782 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝑦𝐵))
91, 8nfxfrd 1497 1 (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1370   = wceq 1372  wnf 1482  wcel 2175  wnfc 2334
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 1469  ax-7 1470  ax-gen 1471  ax-4 1532  ax-17 1548  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-nf 1483  df-cleq 2197  df-nfc 2336
This theorem is referenced by:  nfeld  2363  nfned  2469  vtoclgft  2822  sbcralt  3074  sbcrext  3075  csbiebt  3132  dfnfc2  3867  eusvnfb  4500  eusv2i  4501  iota2df  5256  riota5f  5923
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