Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hbimtg Structured version   Visualization version   GIF version

Theorem hbimtg 30956
 Description: A more general and closed form of hbim 2112. (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 30955 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜒) → (¬ 𝜒 → ∀𝑥 ¬ 𝜑))
2 pm2.21 119 . . . . 5 𝜑 → (𝜑𝜃))
32alimi 1730 . . . 4 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜃))
41, 3syl6 34 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜒) → (¬ 𝜒 → ∀𝑥(𝜑𝜃)))
54adantr 480 . 2 ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → (¬ 𝜒 → ∀𝑥(𝜑𝜃)))
6 ala1 1755 . . . 4 (∀𝑥𝜃 → ∀𝑥(𝜑𝜃))
76imim2i 16 . . 3 ((𝜓 → ∀𝑥𝜃) → (𝜓 → ∀𝑥(𝜑𝜃)))
87adantl 481 . 2 ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → (𝜓 → ∀𝑥(𝜑𝜃)))
95, 8jad 173 1 ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒𝜓) → ∀𝑥(𝜑𝜃)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by:  hbimg  30959
 Copyright terms: Public domain W3C validator