Theorem alanimi 1843

Description: Variant of al2imi 1842 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 417	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1842	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 411	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401

This theorem is referenced by:  19.26  1897  alsyl  1920  ax13  2413  nfeqf  2419  darapti  2717  axextmo  2745  euind  3696  reuind  3725  sbeqalb  3815  bm1.3iiOLD  5267  trin2  6124  ssfi  9156  bj-nnfan  37267  bj-cbv3ta  37309  bj-bm1.3ii  37587  bj-axreprepsep  37599  mpobi123f  38700  mptbi12f  38704  cotrintab  44231  albitr  44964  2alanimi  44973  ichan  48092