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

Theorem nfrmod 3249
Description: Deduction version of nfrmo 3251. (Contributed by NM, 17-Jun-2017.)
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 3056 . 2 (∃*𝑦𝐴 𝜓 ↔ ∃*𝑦(𝑦𝐴𝜓))
2 nfreud.1 . . 3 𝑦𝜑
3 nfcvf 2924 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
43adantl 473 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝑦)
5 nfreud.2 . . . . . 6 (𝜑𝑥𝐴)
65adantr 472 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐴)
74, 6nfeld 2909 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦𝐴)
8 nfreud.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
98adantr 472 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
107, 9nfand 1973 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜓))
112, 10nfmod2 2618 . 2 (𝜑 → Ⅎ𝑥∃*𝑦(𝑦𝐴𝜓))
121, 11nfxfrd 1927 1 (𝜑 → Ⅎ𝑥∃*𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1628  wnf 1855  wcel 2137  ∃*wmo 2606  wnfc 2887  ∃*wrmo 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-eu 2609  df-mo 2610  df-cleq 2751  df-clel 2754  df-nfc 2889  df-rmo 3056
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator