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 413	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1817	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 407	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1539

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 206  df-an 397

This theorem is referenced by:  19.26  1873  alsyl  1896  ax13  2374  nfeqf  2380  darapti  2679  axextmo  2707  vtoclgft  3540  euind  3719  reuind  3748  sbeqalb  3844  bm1.3ii  5301  trin2  6121  ssfi  9169  bj-nnfan  35614  bj-cbv3ta  35652  bj-bm1.3ii  35933  mpobi123f  37018  mptbi12f  37022  cotrintab  42350  albitr  43107  2alanimi  43116  ichan  46109