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Theorem hbimtg 31686
Description: A more general and closed form of hbim 2125. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
hbimtg ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒𝜓) → ∀𝑥(𝜑𝜃)))

Proof of Theorem hbimtg
StepHypRef Expression
1 hbntg 31685 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜒) → (¬ 𝜒 → ∀𝑥 ¬ 𝜑))
2 pm2.21 120 . . . . 5 𝜑 → (𝜑𝜃))
32alimi 1737 . . . 4 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜃))
41, 3syl6 35 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜒) → (¬ 𝜒 → ∀𝑥(𝜑𝜃)))
54adantr 481 . 2 ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → (¬ 𝜒 → ∀𝑥(𝜑𝜃)))
6 ala1 1739 . . . 4 (∀𝑥𝜃 → ∀𝑥(𝜑𝜃))
76imim2i 16 . . 3 ((𝜓 → ∀𝑥𝜃) → (𝜓 → ∀𝑥(𝜑𝜃)))
87adantl 482 . 2 ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → (𝜓 → ∀𝑥(𝜑𝜃)))
95, 8jad 174 1 ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒𝜓) → ∀𝑥(𝜑𝜃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  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
This theorem is referenced by:  hbimg  31689
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