Theorem hbexg 40964

Description: Closed form of nfex 2342. Derived from hbexgVD 41314. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbexg	⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))

Proof of Theorem hbexg

Step	Hyp	Ref	Expression
1		nfa2 2175	. . 3 ⊢ Ⅎ𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)
2		sp 2181	. . . . . . 7 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
3	2	alimi 1811	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
4		nf5 2289	. . . . . 6 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
5	3, 4	sylibr 236	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
6	1, 5	nfexd 2347	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥∃𝑦𝜑)
7		nf5 2289	. . . 4 ⊢ (Ⅎ𝑥∃𝑦𝜑 ↔ ∀𝑥(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))
8	6, 7	sylib 220	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))
9	1, 8	alrimi 2212	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑥(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))
10		alcom 2162	. 2 ⊢ (∀𝑦∀𝑥(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) ↔ ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))
11	9, 10	sylib 220	1 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)