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Theorem nfrmo 3434
Description: Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker nfrmow 3413 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 3380 . 2 (∃*𝑦𝐴 𝜑 ↔ ∃*𝑦(𝑦𝐴𝜑))
2 nftru 1804 . . . 4 𝑦
3 nfcvf 2932 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
4 nfrmo.1 . . . . . . . 8 𝑥𝐴
54a1i 11 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐴)
63, 5nfeld 2917 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝐴)
7 nfrmo.2 . . . . . . 7 𝑥𝜑
87a1i 11 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
96, 8nfand 1897 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦𝐴𝜑))
109adantl 481 . . . 4 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜑))
112, 10nfmod2 2558 . . 3 (⊤ → Ⅎ𝑥∃*𝑦(𝑦𝐴𝜑))
1211mptru 1547 . 2 𝑥∃*𝑦(𝑦𝐴𝜑)
131, 12nfxfr 1853 1 𝑥∃*𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1538  wtru 1541  wnf 1783  wcel 2108  ∃*wmo 2538  wnfc 2890  ∃*wrmo 3379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2540  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rmo 3380
This theorem is referenced by: (None)
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