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Theorem hbimd 2124
Description: Deduction form of bound-variable hypothesis builder hbim 2125. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
Hypotheses
Ref Expression
hbimd.1 (𝜑 → ∀𝑥𝜑)
hbimd.2 (𝜑 → (𝜓 → ∀𝑥𝜓))
hbimd.3 (𝜑 → (𝜒 → ∀𝑥𝜒))
Assertion
Ref Expression
hbimd (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))

Proof of Theorem hbimd
StepHypRef Expression
1 hbimd.1 . . . 4 (𝜑 → ∀𝑥𝜑)
2 hbimd.2 . . . 4 (𝜑 → (𝜓 → ∀𝑥𝜓))
31, 2nf5dh 2024 . . 3 (𝜑 → Ⅎ𝑥𝜓)
4 hbimd.3 . . . 4 (𝜑 → (𝜒 → ∀𝑥𝜒))
51, 4nf5dh 2024 . . 3 (𝜑 → Ⅎ𝑥𝜒)
63, 5nfimd 1821 . 2 (𝜑 → Ⅎ𝑥(𝜓𝜒))
76nf5rd 2064 1 (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-nf 1708
This theorem is referenced by:  dvelimf-o  34033
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