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

Theorem nfrmo 3431
Description: Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2375. Use the weaker nfrmow 3411 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfrmo.1 𝑥𝐴
nfrmo.2 𝑥𝜑
Assertion
Ref Expression
nfrmo 𝑥∃*𝑦𝐴 𝜑

Proof of Theorem nfrmo
StepHypRef Expression
1 df-rmo 3378 . 2 (∃*𝑦𝐴 𝜑 ↔ ∃*𝑦(𝑦𝐴𝜑))
2 nftru 1801 . . . 4 𝑦
3 nfcvf 2930 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
4 nfrmo.1 . . . . . . . 8 𝑥𝐴
54a1i 11 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐴)
63, 5nfeld 2915 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝐴)
7 nfrmo.2 . . . . . . 7 𝑥𝜑
87a1i 11 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
96, 8nfand 1895 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦𝐴𝜑))
109adantl 481 . . . 4 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜑))
112, 10nfmod2 2556 . . 3 (⊤ → Ⅎ𝑥∃*𝑦(𝑦𝐴𝜑))
1211mptru 1544 . 2 𝑥∃*𝑦(𝑦𝐴𝜑)
131, 12nfxfr 1850 1 𝑥∃*𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1535  wtru 1538  wnf 1780  wcel 2106  ∃*wmo 2536  wnfc 2888  ∃*wrmo 3377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rmo 3378
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator