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Theorem hbimtg 33501
Description: A more general and closed form of hbim 2300. (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 33500 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜒) → (¬ 𝜒 → ∀𝑥 ¬ 𝜑))
2 pm2.21 123 . . . . 5 𝜑 → (𝜑𝜃))
32alimi 1819 . . . 4 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜃))
41, 3syl6 35 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜒) → (¬ 𝜒 → ∀𝑥(𝜑𝜃)))
54adantr 484 . 2 ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → (¬ 𝜒 → ∀𝑥(𝜑𝜃)))
6 ala1 1821 . . . 4 (∀𝑥𝜃 → ∀𝑥(𝜑𝜃))
76imim2i 16 . . 3 ((𝜓 → ∀𝑥𝜃) → (𝜓 → ∀𝑥(𝜑𝜃)))
87adantl 485 . 2 ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → (𝜓 → ∀𝑥(𝜑𝜃)))
95, 8jad 190 1 ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒𝜓) → ∀𝑥(𝜑𝜃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788
This theorem is referenced by:  hbimg  33504
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