Theorem bj-nfexd 37633

Description: Variant of nfexd 2362. (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 1801	. 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
2		bj-nfald.1	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-nfald.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	3	nfnd 1879	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
5	2, 4	bj-nfald 37632	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ¬ 𝜓)
6	5	nfnd 1879	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓)
7	1, 6	nfxfrd 1875	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1559  ∃wex 1800  Ⅎwnf 1804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1801  df-nf 1805

This theorem is referenced by:  copsex2d  37636