Theorem alanimi 1734

Description: Variant of al2imi 1733 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 449	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1733	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 444	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385

This theorem is referenced by:  19.26  1786  alsyl  1811  ax13  2237  nfeqf  2289  bm1.1  2595  vtoclgft  3227  vtoclgftOLD  3228  euind  3360  reuind  3378  sbeqalb  3455  bm1.3ii  4712  trin2  5438  bj-cbv3ta  31897  mpt2bi123f  33141  mptbi12f  33145  cotrintab  36940  albitr  37584  2alanimi  37593