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Theorem 19.21ht 1489
Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)
Assertion
Ref Expression
19.21ht (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem 19.21ht
StepHypRef Expression
1 alim 1362 . . . . 5 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
21imim2d 52 . . . 4 (∀𝑥(𝜑𝜓) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜓)))
32com12 30 . . 3 ((𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
43sps 1446 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
5 hba1 1449 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑥(𝜑 → ∀𝑥𝜑))
6 ax-4 1416 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
7 hba1 1449 . . . . 5 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
87a1i 9 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥𝜓 → ∀𝑥𝑥𝜓))
95, 6, 8hbimd 1481 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓)))
10 ax-4 1416 . . . . 5 (∀𝑥𝜓𝜓)
1110imim2i 12 . . . 4 ((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
1211alimi 1360 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
139, 12syl6 33 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓)))
144, 13impbid 124 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  19.21t  1490  sbal2  1914
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