Theorem alanimi 1848

Description: Variant of al2imi 1847 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 1847	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 412	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

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

This theorem is used by:  19.26  1902  alsyl  1925  ax13  2404  nfeqf  2410  darapti  2708  axextmo  2736  euind  3681  reuind  3710  sbeqalb  3800  trin2  6119  ssfi  9196  bj-nnfan  37780  bj-cbv3ta  37822  bj-bm1.3ii  38115  bj-axreprepsep  38127  mpobi123f  39231  mptbi12f  39235  cotrintab  44751  albitr  45484  2alanimi  45493  ichan  48673