Theorem alanimi 1849

Description: Variant of al2imi 1848 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 418	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1848	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 412	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  19.26  1903  alsyl  1926  ax13  2409  nfeqf  2415  darapti  2713  axextmo  2741  euind  3689  reuind  3718  sbeqalb  3808  bm1.3iiOLD  5267  trin2  6125  ssfi  9164  bj-nnfan  37436  bj-cbv3ta  37478  bj-bm1.3ii  37757  bj-axreprepsep  37769  mpobi123f  38869  mptbi12f  38873  cotrintab  44398  albitr  45131  2alanimi  45140  ichan  48262