Theorem nfrmo 3094

Description: Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)

Hypotheses

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

Assertion

Ref	Expression
nfrmo	⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Proof of Theorem nfrmo

Step	Hyp	Ref	Expression
1		df-rmo 2904	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nftru 1721	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfcvf 2774	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4		nfreu.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐴)
6	3, 5	nfeld 2759	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfreu.2	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
8	7	a1i 11	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
9	6, 8	nfand 1814	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	9	adantl 481	. . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
11	2, 10	nfmod2 2471	. . 3 ⊢ (⊤ → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
12	11	trud 1484	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
13	1, 12	nfxfr 1771	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Syntax hints:  ¬ wn 3   ∧ wa 383  ∀wal 1473  ⊤wtru 1476  Ⅎwnf 1699   ∈ wcel 1977  ∃*wmo 2459  Ⅎwnfc 2738  ∃*wrmo 2899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rmo 2904

This theorem is referenced by:  2rmorex  3379  2reurex  39830