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

Theorem alanimi 1911
Description: Variant of al2imi 1910 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
alanimi.1 ((𝜑𝜓) → 𝜒)
Assertion
Ref Expression
alanimi ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Proof of Theorem alanimi
StepHypRef Expression
1 alanimi.1 . . . 4 ((𝜑𝜓) → 𝜒)
21ex 401 . . 3 (𝜑 → (𝜓𝜒))
32al2imi 1910 . 2 (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
43imp 395 1 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904
This theorem depends on definitions:  df-bi 198  df-an 385
This theorem is referenced by:  19.26  1968  alsyl  1991  ax13  2355  nfeqf  2401  darapti  2713  axextmo  2748  bm1.1OLD  2749  vtoclgft  3407  euind  3554  reuind  3574  sbeqalb  3651  bm1.3ii  4946  trin2  5704  bj-cbv3ta  33168  bj-bm1.3ii  33474  mpt2bi123f  34415  mptbi12f  34419  cotrintab  38622  albitr  39262  2alanimi  39271
  Copyright terms: Public domain W3C validator