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Theorem nfreud 3407
Description: Deduction version of nfreu 3409. Usage of this theorem is discouraged because it depends on ax-13 2371. (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 3357 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
2 nfrmod.1 . . 3 𝑦𝜑
3 nfcvf 2937 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
43adantl 483 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
5 nfrmod.2 . . . . . 6 (𝜑𝑥𝐴)
65adantr 482 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
74, 6nfeld 2919 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfrmod.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 482 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfand 1901 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfeud2 2589 . 2 (𝜑 → Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1857 1 (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wal 1540  wnf 1786  wcel 2107  ∃!weu 2567  wnfc 2888  ∃!wreu 3354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-mo 2539  df-eu 2568  df-cleq 2729  df-clel 2815  df-nfc 2890  df-reu 3357
This theorem is referenced by:  nfreu  3409
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