Theorem nfrexdw 3307

Description: Deduction version of nfrexw 3310. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2371. See nfrexd 3369 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 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 3073	. 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 1861	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
6	2, 3, 5	nfraldw 3306	. . 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 1785  Ⅎwnfc 2883  ∀wral 3061  ∃wrex 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071

This theorem is referenced by:  nfrexw  3310  nfunid  4914  nfttrcld  9704  nfiund  47709