Theorem nfrmod 3299

Description: Deduction version of nfrmo 3303. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfreud.1	⊢ Ⅎ𝑦𝜑
nfreud.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfreud.3	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfrmod	⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓)

Proof of Theorem nfrmod

Step	Hyp	Ref	Expression
1		df-rmo 3071	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfreud.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcvf 2935	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4	3	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦)
5		nfreud.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴)
7	4, 6	nfeld 2917	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴)
8		nfreud.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
10	7, 9	nfand 1901	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
11	2, 10	nfmod2 2558	. 2 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
12	1, 11	nfxfrd 1857	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1537  Ⅎwnf 1787   ∈ wcel 2108  ∃*wmo 2538  Ⅎwnfc 2886  ∃*wrmo 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-mo 2540  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rmo 3071

This theorem is referenced by: (None)