Theorem alanimi 1813

Description: Variant of al2imi 1812 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 412	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1812	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 406	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  19.26  1868  alsyl  1891  ax13  2378  nfeqf  2384  darapti  2682  axextmo  2710  euind  3733  reuind  3762  sbeqalb  3859  bm1.3iiOLD  5308  trin2  6146  ssfi  9212  bj-nnfan  36731  bj-cbv3ta  36769  bj-bm1.3ii  37047  mpobi123f  38149  mptbi12f  38153  cotrintab  43604  albitr  44359  2alanimi  44368  ichan  47380