Theorem alanimi 1817

Description: Variant of al2imi 1816 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 415	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1816	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 409	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-an 399

This theorem is referenced by:  19.26  1871  alsyl  1894  ax13  2393  nfeqf  2399  darapti  2769  axextmo  2797  vtoclgft  3553  vtoclgftOLD  3554  euind  3715  reuind  3744  sbeqalb  3836  bm1.3ii  5206  trin2  5983  bj-nnfan  34077  bj-cbv3ta  34108  bj-bm1.3ii  34360  mpobi123f  35455  mptbi12f  35459  cotrintab  39994  albitr  40715  2alanimi  40724