Theorem nfrexdg 3227

Description: Deduction version of nfrexg 3229. Usage of this theorem is discouraged because it depends on ax-13 2371. See nfrexd 3226 for a version with a disjoint variable condition, but not requiring ax-13 2371. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem nfrexdg

Step	Hyp	Ref	Expression
1		dfrex2 3161	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
2		nfrexdg.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfrexdg.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4		nfrexdg.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	4	nfnd 1866	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
6	2, 3, 5	nfrald 3146	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓)
7	6	nfnd 1866	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
8	1, 7	nfxfrd 1861	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1791  Ⅎwnfc 2884  ∀wral 3061  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067

This theorem is referenced by:  nfrexg  3229  nfiundg  46052