Theorem bj-nfexd 38249

Description: Variant of nfexd 2363. (Contributed by BJ, 25-Dec-2023.)

Hypotheses

Ref	Expression
bj-nfald.1	⊢ (𝜑 → ∀𝑦𝜑)
bj-nfald.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
bj-nfexd	⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)

Proof of Theorem bj-nfexd

Step	Hyp	Ref	Expression
1		df-ex 1813	. 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
2		bj-nfald.1	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-nfald.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	3	nfnd 1891	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
5	2, 4	bj-nfald 38248	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ¬ 𝜓)
6	5	nfnd 1891	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓)
7	1, 6	nfxfrd 1887	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  copsex2d  38252