MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfrmod Structured version   Visualization version   GIF version

Theorem nfrmod 3364
Description: Deduction version of nfrmo 3368. Usage of this theorem is discouraged because it depends on ax-13 2392. (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 3140 . 2 (∃*𝑦𝐴 𝜓 ↔ ∃*𝑦(𝑦𝐴𝜓))
2 nfreud.1 . . 3 𝑦𝜑
3 nfcvf 3007 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
43adantl 485 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
5 nfreud.2 . . . . . 6 (𝜑𝑥𝐴)
65adantr 484 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
74, 6nfeld 2991 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfreud.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 484 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfand 1899 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfmod2 2643 . 2 (𝜑 → Ⅎ𝑥∃*𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1855 1 (𝜑 → Ⅎ𝑥∃*𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1536  wnf 1785  wcel 2115  ∃*wmo 2622  wnfc 2962  ∃*wrmo 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2624  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rmo 3140
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator