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Theorem nfrmod 3301
Description: Deduction version of nfrmo 3307. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfreud.1 𝑦𝜑
nfreud.2 (𝜑𝑥𝐴)
nfreud.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrmod (𝜑 → Ⅎ𝑥∃*𝑦𝐴 𝜓)

Proof of Theorem nfrmod
StepHypRef Expression
1 df-rmo 3072 . 2 (∃*𝑦𝐴 𝜓 ↔ ∃*𝑦(𝑦𝐴𝜓))
2 nfreud.1 . . 3 𝑦𝜑
3 nfcvf 2936 . . . . . 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 2918 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfreud.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 481 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfand 1900 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfmod2 2558 . 2 (𝜑 → Ⅎ𝑥∃*𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1856 1 (𝜑 → Ⅎ𝑥∃*𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1537  wnf 1786  wcel 2106  ∃*wmo 2538  wnfc 2887  ∃*wrmo 3067
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-mo 2540  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rmo 3072
This theorem is referenced by: (None)
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