MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alrimdh Structured version   Visualization version   GIF version

Theorem alrimdh 1855
Description: Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2197 and 19.21h 2286. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
Hypotheses
Ref Expression
alrimdh.1 (𝜑 → ∀𝑥𝜑)
alrimdh.2 (𝜓 → ∀𝑥𝜓)
alrimdh.3 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
alrimdh (𝜑 → (𝜓 → ∀𝑥𝜒))

Proof of Theorem alrimdh
StepHypRef Expression
1 alrimdh.2 . 2 (𝜓 → ∀𝑥𝜓)
2 alrimdh.1 . . 3 (𝜑 → ∀𝑥𝜑)
3 alrimdh.3 . . 3 (𝜑 → (𝜓𝜒))
42, 3alimdh 1809 . 2 (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
51, 4syl5 34 1 (𝜑 → (𝜓 → ∀𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1787  ax-4 1801
This theorem is referenced by:  alrimdv  1921  ax12indn  35959  gen21nv  40831
  Copyright terms: Public domain W3C validator