Theorem nfexd2 1973

Description: Variation on nfexd 1854 which adds the hypothesis that x and y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)

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 1542	. 2 ⊢ (∃yψ ↔ ¬ ∀y ¬ ψ)
2		nfald2.1	. . . 4 ⊢ Ⅎyφ
3		nfald2.2	. . . . 5 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)
4	3	nfnd 1791	. . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx ¬ ψ)
5	2, 4	nfald2 1972	. . 3 ⊢ (φ → Ⅎx∀y ¬ ψ)
6	5	nfnd 1791	. 2 ⊢ (φ → Ⅎx ¬ ∀y ¬ ψ)
7	1, 6	nfxfrd 1571	1 ⊢ (φ → Ⅎx∃yψ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is used by:  nfeud2  2216  nfmod2  2217