Theorem alanimi 1816

Description: Variant of al2imi 1815 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)

Hypothesis

Ref	Expression
alanimi.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
alanimi	⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Proof of Theorem alanimi

Step	Hyp	Ref	Expression
1		alanimi.1	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	ex 411	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1815	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 405	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 206  df-an 395

This theorem is referenced by:  19.26  1871  alsyl  1894  ax13  2372  nfeqf  2378  darapti  2677  axextmo  2705  vtoclgft  3539  euind  3719  reuind  3748  sbeqalb  3844  bm1.3ii  5301  trin2  6123  ssfi  9175  bj-nnfan  35929  bj-cbv3ta  35967  bj-bm1.3ii  36248  mpobi123f  37333  mptbi12f  37337  cotrintab  42667  albitr  43424  2alanimi  43433  ichan  46421