Theorem nfrmo 3417

Description: Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2375. Use the weaker nfrmow 3396 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 3363	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nftru 1803	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfcvf 2924	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4		nfrmo.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐴)
6	3, 5	nfeld 2909	. . . . . 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 1896	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	9	adantl 481	. . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
11	2, 10	nfmod2 2556	. . 3 ⊢ (⊤ → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
12	11	mptru 1546	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
13	1, 12	nfxfr 1852	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1537  ⊤wtru 1540  Ⅎwnf 1782   ∈ wcel 2107  ∃*wmo 2536  Ⅎwnfc 2882  ∃*wrmo 3362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2375  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-mo 2538  df-cleq 2726  df-clel 2808  df-nfc 2884  df-rmo 3363

This theorem is referenced by: (None)