Theorem nfrexd 3347

Description: Deduction version of nfrex 3349. Usage of this theorem is discouraged because it depends on ax-13 2370. See nfrexdw 3284 for a version with a disjoint variable condition, but not requiring ax-13 2370. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem nfrexd

Step	Hyp	Ref	Expression
1		dfrex2 3056	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
2		nfrald.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfrald.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4		nfrald.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	4	nfnd 1858	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
6	2, 3, 5	nfrald 3346	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓)
7	6	nfnd 1858	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
8	1, 7	nfxfrd 1854	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1783  Ⅎwnfc 2876  ∀wral 3044  ∃wrex 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2370  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054

This theorem is referenced by:  nfrex  3349  nfiundg  49661