Theorem nfrexdg 3241

Description: Deduction version of nfrexg 3243. Usage of this theorem is discouraged because it depends on ax-13 2372. See nfrexd 3240 for a version with a disjoint variable condition, but not requiring ax-13 2372. (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 3170	. 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 1861	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
6	2, 3, 5	nfrald 3150	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓)
7	6	nfnd 1861	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
8	1, 7	nfxfrd 1856	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1786  Ⅎwnfc 2887  ∀wral 3064  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070

This theorem is referenced by:  nfrexg  3243  nfiundg  46381