Theorem alanimi 1819

Description: Variant of al2imi 1818 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 1818	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 407	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397

This theorem is referenced by:  19.26  1873  alsyl  1896  ax13  2375  nfeqf  2381  darapti  2685  axextmo  2713  vtoclgft  3492  euind  3659  reuind  3688  sbeqalb  3784  bm1.3ii  5226  trin2  6028  ssfi  8956  bj-nnfan  34930  bj-cbv3ta  34968  bj-bm1.3ii  35235  mpobi123f  36320  mptbi12f  36324  cotrintab  41222  albitr  41981  2alanimi  41990  ichan  44907