Theorem nfrmod 3373

Description: Deduction version of nfrmo 3377. Usage of this theorem is discouraged because it depends on ax-13 2386. (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 3146	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfreud.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcvf 3007	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4	3	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦)
5		nfreud.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	5	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴)
7	4, 6	nfeld 2989	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴)
8		nfreud.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
9	8	adantr 483	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
10	7, 9	nfand 1894	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
11	2, 10	nfmod2 2638	. 2 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
12	1, 11	nfxfrd 1850	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1531  Ⅎwnf 1780   ∈ wcel 2110  ∃*wmo 2616  Ⅎwnfc 2961  ∃*wrmo 3141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-mo 2618  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rmo 3146

This theorem is referenced by: (None)