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 416	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1817	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 410	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536

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 210  df-an 400

This theorem is referenced by:  19.26  1871  alsyl  1894  ax13  2382  nfeqf  2388  darapti  2746  axextmo  2774  vtoclgft  3501  vtoclgftOLD  3502  euind  3663  reuind  3692  sbeqalb  3783  bm1.3ii  5170  trin2  5950  bj-nnfan  34192  bj-cbv3ta  34223  bj-bm1.3ii  34481  mpobi123f  35600  mptbi12f  35604  cotrintab  40314  albitr  41067  2alanimi  41076  ichan  43972