Theorem alanimi 1818

Description: Variant of al2imi 1817 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 412	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1817	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 406	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  19.26  1872  alsyl  1895  ax13  2380  nfeqf  2386  darapti  2685  axextmo  2713  euind  3671  reuind  3700  sbeqalb  3792  bm1.3iiOLD  5238  trin2  6081  ssfi  9101  bj-nnfan  37070  bj-cbv3ta  37112  bj-bm1.3ii  37390  bj-axreprepsep  37401  mpobi123f  38500  mptbi12f  38504  cotrintab  44062  albitr  44811  2alanimi  44820  ichan  47930