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Theorem nfimdOLDOLD 1821
Description: Obsolete proof of nfimd 1820 as of 3-Nov-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1707 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfimd.1 (𝜑 → Ⅎ𝑥𝜓)
nfimd.2 (𝜑 → Ⅎ𝑥𝜒)
Assertion
Ref Expression
nfimdOLDOLD (𝜑 → Ⅎ𝑥(𝜓𝜒))

Proof of Theorem nfimdOLDOLD
StepHypRef Expression
1 19.35 1802 . . 3 (∃𝑥(𝜓𝜒) ↔ (∀𝑥𝜓 → ∃𝑥𝜒))
2 nfimd.1 . . . . . 6 (𝜑 → Ⅎ𝑥𝜓)
3 nf4 1710 . . . . . 6 (Ⅎ𝑥𝜓 ↔ (¬ ∀𝑥𝜓 → ∀𝑥 ¬ 𝜓))
42, 3sylib 208 . . . . 5 (𝜑 → (¬ ∀𝑥𝜓 → ∀𝑥 ¬ 𝜓))
5 pm2.21 120 . . . . . 6 𝜓 → (𝜓𝜒))
65alimi 1736 . . . . 5 (∀𝑥 ¬ 𝜓 → ∀𝑥(𝜓𝜒))
74, 6syl6 35 . . . 4 (𝜑 → (¬ ∀𝑥𝜓 → ∀𝑥(𝜓𝜒)))
8 nfimd.2 . . . . . 6 (𝜑 → Ⅎ𝑥𝜒)
98nfrd 1714 . . . . 5 (𝜑 → (∃𝑥𝜒 → ∀𝑥𝜒))
10 ala1 1738 . . . . 5 (∀𝑥𝜒 → ∀𝑥(𝜓𝜒))
119, 10syl6 35 . . . 4 (𝜑 → (∃𝑥𝜒 → ∀𝑥(𝜓𝜒)))
127, 11jad 174 . . 3 (𝜑 → ((∀𝑥𝜓 → ∃𝑥𝜒) → ∀𝑥(𝜓𝜒)))
131, 12syl5bi 232 . 2 (𝜑 → (∃𝑥(𝜓𝜒) → ∀𝑥(𝜓𝜒)))
1413nfd 1713 1 (𝜑 → Ⅎ𝑥(𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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