Theorem alanimi 1836

Description: Variant of al2imi 1835 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 1835	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 410	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829

This theorem depends on definitions:  df-bi 209  df-an 400

This theorem is referenced by:  19.26  1890  alsyl  1913  ax13  2406  nfeqf  2412  darapti  2710  axextmo  2738  euind  3687  reuind  3716  sbeqalb  3806  bm1.3iiOLD  5252  trin2  6110  ssfi  9141  bj-nnfan  37229  bj-cbv3ta  37271  bj-bm1.3ii  37549  bj-axreprepsep  37560  mpobi123f  38661  mptbi12f  38665  cotrintab  44190  albitr  44939  2alanimi  44948  ichan  48061