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Theorem nfreud 3370
Description: Deduction version of nfreu 3374. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfreud.1 𝑦𝜑
nfreud.2 (𝜑𝑥𝐴)
nfreud.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfreud (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)

Proof of Theorem nfreud
StepHypRef Expression
1 df-reu 3142 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
2 nfreud.1 . . 3 𝑦𝜑
3 nfcvf 3004 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
43adantl 482 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
5 nfreud.2 . . . . . 6 (𝜑𝑥𝐴)
65adantr 481 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
74, 6nfeld 2986 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfreud.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 481 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfand 1889 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfeud2 2669 . 2 (𝜑 → Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1845 1 (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1526  wnf 1775  wcel 2105  ∃!weu 2646  wnfc 2958  ∃!wreu 3137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-mo 2615  df-eu 2647  df-cleq 2811  df-clel 2890  df-nfc 2960  df-reu 3142
This theorem is referenced by:  nfreu  3374
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