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

Theorem nfrmow 3362
Description: Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3364 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
nfreuw.1 𝑥𝐴
nfreuw.2 𝑥𝜑
Assertion
Ref Expression
nfrmow 𝑥∃*𝑦𝐴 𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrmow
StepHypRef Expression
1 df-rmo 3133 . 2 (∃*𝑦𝐴 𝜑 ↔ ∃*𝑦(𝑦𝐴𝜑))
2 nftru 1805 . . . 4 𝑦
3 nfcvd 2974 . . . . . 6 (⊤ → 𝑥𝑦)
4 nfreuw.1 . . . . . . 7 𝑥𝐴
54a1i 11 . . . . . 6 (⊤ → 𝑥𝐴)
63, 5nfeld 2984 . . . . 5 (⊤ → Ⅎ𝑥 𝑦𝐴)
7 nfreuw.2 . . . . . 6 𝑥𝜑
87a1i 11 . . . . 5 (⊤ → Ⅎ𝑥𝜑)
96, 8nfand 1898 . . . 4 (⊤ → Ⅎ𝑥(𝑦𝐴𝜑))
102, 9nfmodv 2642 . . 3 (⊤ → Ⅎ𝑥∃*𝑦(𝑦𝐴𝜑))
1110mptru 1544 . 2 𝑥∃*𝑦(𝑦𝐴𝜑)
121, 11nfxfr 1853 1 𝑥∃*𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  wa 398  wtru 1538  wnf 1784  wcel 2114  ∃*wmo 2620  wnfc 2957  ∃*wrmo 3128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-mo 2622  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rmo 3133
This theorem is referenced by:  2rmorex  3725  2reurex  3730
  Copyright terms: Public domain W3C validator