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Theorem nfalt 1540
Description: Closed form of nfal 1538. (Contributed by Jim Kingdon, 11-May-2018.)
Assertion
Ref Expression
nfalt (∀𝑦𝑥𝜑 → Ⅎ𝑥𝑦𝜑)

Proof of Theorem nfalt
StepHypRef Expression
1 alim 1416 . . . 4 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦𝑥𝜑))
2 alcom 1437 . . . 4 (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑)
31, 2syl6ib 160 . . 3 (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))
43alimi 1414 . 2 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(∀𝑦𝜑 → ∀𝑥𝑦𝜑))
5 df-nf 1420 . . . 4 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
65albii 1429 . . 3 (∀𝑦𝑥𝜑 ↔ ∀𝑦𝑥(𝜑 → ∀𝑥𝜑))
7 alcom 1437 . . 3 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝑦(𝜑 → ∀𝑥𝜑))
86, 7bitri 183 . 2 (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 → ∀𝑥𝜑))
9 df-nf 1420 . 2 (Ⅎ𝑥𝑦𝜑 ↔ ∀𝑥(∀𝑦𝜑 → ∀𝑥𝑦𝜑))
104, 8, 93imtr4i 200 1 (∀𝑦𝑥𝜑 → Ⅎ𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1312  wnf 1419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408
This theorem depends on definitions:  df-bi 116  df-nf 1420
This theorem is referenced by:  dvelimor  1969
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