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Theorem hbexg 41259
Description: Closed form of nfex 2335. Derived from hbexgVD 41609. (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 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Proof of Theorem hbexg
StepHypRef Expression
1 nfa2 2175 . . 3 𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑)
2 sp 2181 . . . . . . 7 (∀𝑦(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
32alimi 1813 . . . . . 6 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
4 nf5 2288 . . . . . 6 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
53, 4sylibr 237 . . . . 5 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
61, 5nfexd 2340 . . . 4 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝑦𝜑)
7 nf5 2288 . . . 4 (Ⅎ𝑥𝑦𝜑 ↔ ∀𝑥(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
86, 7sylib 221 . . 3 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
91, 8alrimi 2212 . 2 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑥(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
10 alcom 2161 . 2 (∀𝑦𝑥(∃𝑦𝜑 → ∀𝑥𝑦𝜑) ↔ ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
119, 10sylib 221 1 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by: (None)
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