Theorem alanimi 1823

Description: Variant of al2imi 1822 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 413	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1822	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 407	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 208  df-an 397

This theorem is referenced by:  19.26  1877  alsyl  1900  ax13  2383  nfeqf  2389  darapti  2687  axextmo  2715  euind  3665  reuind  3694  sbeqalb  3785  bm1.3iiOLD  5224  trin2  6073  ssfi  9097  bj-nnfan  37097  bj-cbv3ta  37139  bj-bm1.3ii  37417  bj-axreprepsep  37428  mpobi123f  38529  mptbi12f  38533  cotrintab  44058  albitr  44807  2alanimi  44816  ichan  47930