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Theorem nfimdOLD 2262
 Description: Obsolete proof of nfimd 1863 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfimdOLD.1 (𝜑 → Ⅎ𝑥𝜓)
nfimdOLD.2 (𝜑 → Ⅎ𝑥𝜒)
Assertion
Ref Expression
nfimdOLD (𝜑 → Ⅎ𝑥(𝜓𝜒))

Proof of Theorem nfimdOLD
StepHypRef Expression
1 nfimdOLD.1 . 2 (𝜑 → Ⅎ𝑥𝜓)
2 nfimdOLD.2 . 2 (𝜑 → Ⅎ𝑥𝜒)
3 nfnf1OLDOLD 2244 . . . 4 𝑥𝑥𝜓
4 nfnf1OLDOLD 2244 . . . 4 𝑥𝑥𝜒
5 nfrOLD 2224 . . . . . 6 (Ⅎ𝑥𝜒 → (𝜒 → ∀𝑥𝜒))
65imim2d 57 . . . . 5 (Ⅎ𝑥𝜒 → ((𝜓𝜒) → (𝜓 → ∀𝑥𝜒)))
7 19.21tOLD 2249 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥(𝜓𝜒) ↔ (𝜓 → ∀𝑥𝜒)))
87biimprd 238 . . . . 5 (Ⅎ𝑥𝜓 → ((𝜓 → ∀𝑥𝜒) → ∀𝑥(𝜓𝜒)))
96, 8syl9r 78 . . . 4 (Ⅎ𝑥𝜓 → (Ⅎ𝑥𝜒 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒))))
103, 4, 9alrimdOLD 2232 . . 3 (Ⅎ𝑥𝜓 → (Ⅎ𝑥𝜒 → ∀𝑥((𝜓𝜒) → ∀𝑥(𝜓𝜒))))
11 df-nfOLD 1761 . . 3 (Ⅎ𝑥(𝜓𝜒) ↔ ∀𝑥((𝜓𝜒) → ∀𝑥(𝜓𝜒)))
1210, 11syl6ibr 242 . 2 (Ⅎ𝑥𝜓 → (Ⅎ𝑥𝜒 → Ⅎ𝑥(𝜓𝜒)))
131, 2, 12sylc 65 1 (𝜑 → Ⅎ𝑥(𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521  ℲwnfOLD 1749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087 This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1745  df-nf 1750  df-nfOLD 1761 This theorem is referenced by:  hbimdOLD  2266  nfandOLD  2268  nfbidOLD  2278
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