Theorem nfrexdg 3307

Description: Deduction version of nfrexg 3309. Usage of this theorem is discouraged because it depends on ax-13 2389. See nfrexd 3306 for a version with a disjoint variable condition, but not requiring ax-13 2389. (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 3238	. 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 1857	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
6	2, 3, 5	nfrald 3223	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓)
7	6	nfnd 1857	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
8	1, 7	nfxfrd 1853	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1783  Ⅎwnfc 2960  ∀wral 3137  ∃wrex 3138

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143

This theorem is referenced by:  nfrexg  3309  nfiundg  44848