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Theorem nfrmo 3105
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 2915 . 2 (∃*𝑦𝐴 𝜑 ↔ ∃*𝑦(𝑦𝐴𝜑))
2 nftru 1727 . . . 4 𝑦
3 nfcvf 2784 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
4 nfreu.1 . . . . . . . 8 𝑥𝐴
54a1i 11 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐴)
63, 5nfeld 2769 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝐴)
7 nfreu.2 . . . . . . 7 𝑥𝜑
87a1i 11 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
96, 8nfand 1823 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦𝐴𝜑))
109adantl 482 . . . 4 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜑))
112, 10nfmod2 2482 . . 3 (⊤ → Ⅎ𝑥∃*𝑦(𝑦𝐴𝜑))
1211trud 1490 . 2 𝑥∃*𝑦(𝑦𝐴𝜑)
131, 12nfxfr 1776 1 𝑥∃*𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384  wal 1478  wtru 1481  wnf 1705  wcel 1987  ∃*wmo 2470  wnfc 2748  ∃*wrmo 2910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-eu 2473  df-mo 2474  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rmo 2915
This theorem is referenced by:  2rmorex  3394  2reurex  40482
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