Theorem bj-nfext 37142

Description: Closed form of nfex 2355. (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 2315	. . . . 5 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
2	1	biimpi 218	. . . 4 ⊢ (Ⅎ𝑦𝜑 → ∀𝑦(𝜑 → ∀𝑦𝜑))
3	2	alimi 1830	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑))
4		nfa2 2208	. . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑)
5		bj-hbext 37139	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
6	4, 5	alrimi 2247	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
7	3, 6	syl 17	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
8		nf5 2315	. 2 ⊢ (Ⅎ𝑦∃𝑥𝜑 ↔ ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
9	7, 8	sylibr 236	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1557  ∃wex 1798  Ⅎwnf 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-11 2190  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1799  df-nf 1803

This theorem is referenced by: (None)