Theorem nfrmowOLD 3410

Description: Obsolete version of nfrmow 3397 as of 21-Nov-2024. (Contributed by NM, 16-Jun-2017.) (Revised by GG, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nfrmowOLD	⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrmowOLD

Step	Hyp	Ref	Expression
1		df-rmo 3364	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2		nftru 1799	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfcvd 2893	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝑦)
4		nfreuwOLD.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝐴)
6	3, 5	nfeld 2904	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfreuwOLD.2	. . . . . 6 ⊢ Ⅎ𝑥𝜑
8	7	a1i 11	. . . . 5 ⊢ (⊤ → Ⅎ𝑥𝜑)
9	6, 8	nfand 1893	. . . 4 ⊢ (⊤ → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	2, 9	nfmodv 2548	. . 3 ⊢ (⊤ → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
11	10	mptru 1541	. 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)
12	1, 11	nfxfr 1848	1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 394  ⊤wtru 1535  Ⅎwnf 1778   ∈ wcel 2099  ∃*wmo 2527  Ⅎwnfc 2876  ∃*wrmo 3363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-mo 2529  df-cleq 2718  df-clel 2803  df-nfc 2878  df-rmo 3364

This theorem is referenced by: (None)