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Theorem nfrmo 3249
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 3054 . 2 (∃*𝑦𝐴 𝜑 ↔ ∃*𝑦(𝑦𝐴𝜑))
2 nftru 1875 . . . 4 𝑦
3 nfcvf 2922 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
4 nfreu.1 . . . . . . . 8 𝑥𝐴
54a1i 11 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐴)
63, 5nfeld 2907 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝐴)
7 nfreu.2 . . . . . . 7 𝑥𝜑
87a1i 11 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
96, 8nfand 1971 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦𝐴𝜑))
109adantl 473 . . . 4 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜑))
112, 10nfmod2 2616 . . 3 (⊤ → Ⅎ𝑥∃*𝑦(𝑦𝐴𝜑))
1211trud 1638 . 2 𝑥∃*𝑦(𝑦𝐴𝜑)
131, 12nfxfr 1924 1 𝑥∃*𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383  wal 1626  wtru 1629  wnf 1853  wcel 2135  ∃*wmo 2604  wnfc 2885  ∃*wrmo 3049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-eu 2607  df-mo 2608  df-cleq 2749  df-clel 2752  df-nfc 2887  df-rmo 3054
This theorem is referenced by:  2rmorex  3549  2reurex  41683
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