Theorem nfexd2 2449

Description: Variation on nfexd 2328 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2375. Use nfexd 2328 instead. (New usage is discouraged.)

Hypotheses

Ref	Expression
nfald2.1	⊢ Ⅎ𝑦𝜑
nfald2.2	⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfexd2	⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)

Proof of Theorem nfexd2

Step	Hyp	Ref	Expression
1		df-ex 1777	. 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
2		nfald2.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfald2.2	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
4	3	nfnd 1856	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 ¬ 𝜓)
5	2, 4	nfald2 2448	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ¬ 𝜓)
6	5	nfnd 1856	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓)
7	1, 6	nfxfrd 1851	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1535  ∃wex 1776  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781

This theorem is referenced by:  nfeud2  2588