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Theorem nfreud 3426
Description: Deduction version of nfreu 3428. Usage of this theorem is discouraged because it depends on ax-13 2367. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfrmod.1 𝑦𝜑
nfrmod.2 (𝜑𝑥𝐴)
nfrmod.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfreud (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)

Proof of Theorem nfreud
StepHypRef Expression
1 df-reu 3374 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
2 nfrmod.1 . . 3 𝑦𝜑
3 nfcvf 2929 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
43adantl 481 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
5 nfrmod.2 . . . . . 6 (𝜑𝑥𝐴)
65adantr 480 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
74, 6nfeld 2911 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfrmod.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 480 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfand 1893 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfeud2 2580 . 2 (𝜑 → Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1849 1 (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1532  wnf 1778  wcel 2099  ∃!weu 2558  wnfc 2879  ∃!wreu 3371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2367  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-mo 2530  df-eu 2559  df-cleq 2720  df-clel 2806  df-nfc 2881  df-reu 3374
This theorem is referenced by:  nfreu  3428
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