Theorem alanimi 1846

Description: Variant of al2imi 1845 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 1845	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 411	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839

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

This theorem is referenced by:  19.26  1900  alsyl  1923  ax13  2407  nfeqf  2413  darapti  2711  axextmo  2739  euind  3687  reuind  3716  sbeqalb  3806  bm1.3iiOLD  5265  trin2  6123  ssfi  9153  bj-nnfan  37379  bj-cbv3ta  37421  bj-bm1.3ii  37700  bj-axreprepsep  37712  mpobi123f  38811  mptbi12f  38815  cotrintab  44340  albitr  45073  2alanimi  45082  ichan  48204