Theorem alanimi 1816

Description: Variant of al2imi 1815 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 1815	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 406	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

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

This theorem is referenced by:  19.26  1870  alsyl  1893  ax13  2379  nfeqf  2385  darapti  2683  axextmo  2711  euind  3707  reuind  3736  sbeqalb  3828  bm1.3iiOLD  5272  trin2  6112  ssfi  9187  bj-nnfan  36766  bj-cbv3ta  36804  bj-bm1.3ii  37082  mpobi123f  38186  mptbi12f  38190  cotrintab  43638  albitr  44387  2alanimi  44396  ichan  47469