Theorem bj-nfext 37398

Description: Closed form of nfex 2359. (Contributed by BJ, 10-Oct-2019.)

Assertion

Ref	Expression
bj-nfext	⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)

Proof of Theorem bj-nfext

Step	Hyp	Ref	Expression
1		nf5 2319	. . . . 5 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
2	1	biimpi 219	. . . 4 ⊢ (Ⅎ𝑦𝜑 → ∀𝑦(𝜑 → ∀𝑦𝜑))
3	2	alimi 1844	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑))
4		nfa2 2213	. . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑)
5		bj-hbext 37395	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
6	4, 5	alrimi 2252	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
7	3, 6	syl 18	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
8		nf5 2319	. 2 ⊢ (Ⅎ𝑦∃𝑥𝜑 ↔ ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
9	7, 8	sylibr 237	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)

This proof depends on syntax axioms:   → 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 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  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: (None)