Theorem alanimi 1744

Description: Variant of al2imi 1743 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 450	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1743	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 445	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by:  19.26  1798  alsyl  1820  ax13  2249  nfeqf  2301  bm1.1  2607  vtoclgft  3254  vtoclgftOLD  3255  euind  3393  reuind  3411  sbeqalb  3488  bm1.3ii  4784  trin2  5519  bj-cbv3ta  32710  mpt2bi123f  33971  mptbi12f  33975  cotrintab  37921  albitr  38562  2alanimi  38571