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

Theorem alimdh 1785
Description: Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1778. (Contributed by NM, 4-Jan-2002.)
Hypotheses
Ref Expression
alimdh.1 (𝜑 → ∀𝑥𝜑)
alimdh.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
alimdh (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Proof of Theorem alimdh
StepHypRef Expression
1 alimdh.1 . 2 (𝜑 → ∀𝑥𝜑)
2 alimdh.2 . . 3 (𝜑 → (𝜓𝜒))
32al2imi 1783 . 2 (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
41, 3syl 17 1 (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1762  ax-4 1777
This theorem is referenced by:  alrimdh  1830  alimdv  1885  hbald  2081  alimd  2119  alimdOLD  2227  dral1-o  34508  ax12indalem  34549  ax12inda2ALT  34550
  Copyright terms: Public domain W3C validator