Theorem nfrmo 3414

Description: Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2405. Use the weaker nfrmow 3398 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfrmo.1	⊢ Ⅎ𝑥𝐴
nfrmo.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrmo	⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Proof of Theorem nfrmo

Step	Hyp	Ref	Expression
1		df-rmo 3369	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nftru 1826	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfcvf 2952	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4		nfrmo.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐴)
6	3, 5	nfeld 2937	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfrmo.2	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
8	7	a1i 11	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
9	6, 8	nfand 1919	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	9	adantl 485	. . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
11	2, 10	nfmod2 2587	. . 3 ⊢ (⊤ → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
12	11	mptru 1569	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
13	1, 12	nfxfr 1875	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Syntax hints:  ¬ wn 3   ∧ wa 399  ∀wal 1560  ⊤wtru 1563  Ⅎwnf 1805   ∈ wcel 2144  ∃*wmo 2566  Ⅎwnfc 2911  ∃*wrmo 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-13 2405  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-mo 2568  df-cleq 2756  df-clel 2839  df-nfc 2913  df-rmo 3369

This theorem is referenced by: (None)