Theorem alanimi 1817

Description: Variant of al2imi 1816 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 1816	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 406	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 206  df-an 396

This theorem is referenced by:  19.26  1872  alsyl  1895  ax13  2373  nfeqf  2379  darapti  2678  axextmo  2706  vtoclgft  3540  euind  3720  reuind  3749  sbeqalb  3845  bm1.3ii  5302  trin2  6124  ssfi  9176  bj-nnfan  35930  bj-cbv3ta  35968  bj-bm1.3ii  36249  mpobi123f  37334  mptbi12f  37338  cotrintab  42668  albitr  43425  2alanimi  43434  ichan  46422