Theorem nfrexd 3247

Description: Deduction version of nfrex 3248. (Contributed by Mario Carneiro, 14-Oct-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem nfrexd

Step	Hyp	Ref	Expression
1		dfrex2 3181	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
2		nfrexd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfrexd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4		nfrexd.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	4	nfnd 1821	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
6	2, 3, 5	nfrald 3168	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓)
7	6	nfnd 1821	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
8	1, 7	nfxfrd 1817	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1747  Ⅎwnfc 2911  ∀wral 3083  ∃wrex 3084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089

This theorem is referenced by:  nfrex  3248  nfunid  4716  nfiund  44175