Theorem nfrexdw 3310

Description: Deduction version of nfrexw 3312. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2403. See nfrexd 3360 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)   𝐴(𝑥, 𝑦)

Proof of Theorem nfrexdw

Step	Hyp	Ref	Expression
1		dfrex2 3091	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
2		nfraldw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfraldw.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4		nfraldw.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	4	nfnd 1891	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
6	2, 3, 5	nfraldw 3309	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓)
7	6	nfnd 1891	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
8	1, 7	nfxfrd 1887	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  Ⅎwnf 1816  Ⅎwnfc 2909  ∀wral 3078  ∃wrex 3088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-10 2178  ax-11 2194  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089

This theorem is used by:  nfrexw  3312  nfunid  4876  nfttrcld  9692  nfchnd  18701  nfiund  50585