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

Theorem nfrmo 3217
Description: Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
nfreu.1 𝑥𝐴
nfreu.2 𝑥𝜑
Assertion
Ref Expression
nfrmo 𝑥∃*𝑦𝐴 𝜑

Proof of Theorem nfrmo
StepHypRef Expression
1 df-rmo 3022 . 2 (∃*𝑦𝐴 𝜑 ↔ ∃*𝑦(𝑦𝐴𝜑))
2 nftru 1843 . . . 4 𝑦
3 nfcvf 2890 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
4 nfreu.1 . . . . . . . 8 𝑥𝐴
54a1i 11 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐴)
63, 5nfeld 2875 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝐴)
7 nfreu.2 . . . . . . 7 𝑥𝜑
87a1i 11 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
96, 8nfand 1939 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦𝐴𝜑))
109adantl 473 . . . 4 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜑))
112, 10nfmod2 2584 . . 3 (⊤ → Ⅎ𝑥∃*𝑦(𝑦𝐴𝜑))
1211trud 1606 . 2 𝑥∃*𝑦(𝑦𝐴𝜑)
131, 12nfxfr 1892 1 𝑥∃*𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383  wal 1594  wtru 1597  wnf 1821  wcel 2103  ∃*wmo 2572  wnfc 2853  ∃*wrmo 3017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-eu 2575  df-mo 2576  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rmo 3022
This theorem is referenced by:  2rmorex  3518  2reurex  41604
  Copyright terms: Public domain W3C validator