Theorem bj-nfext 34048

Description: Closed form of nfex 2343. (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 2290	. . . . 5 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
2	1	biimpi 218	. . . 4 ⊢ (Ⅎ𝑦𝜑 → ∀𝑦(𝜑 → ∀𝑦𝜑))
3	2	alimi 1812	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑))
4		nfa2 2176	. . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑)
5		bj-hbext 34046	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
6	4, 5	alrimi 2213	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
7	3, 6	syl 17	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
8		nf5 2290	. 2 ⊢ (Ⅎ𝑦∃𝑥𝜑 ↔ ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
9	7, 8	sylibr 236	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)