Theorem alanimi 1562

Description: Variant of al2imi 1561 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 423	. . 3 ⊢ (φ → (ψ → χ))
3	2	al2imi 1561	. 2 ⊢ (∀xφ → (∀xψ → ∀xχ))
4	3	imp 418	1 ⊢ ((∀xφ ∧ ∀xψ) → ∀xχ)

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  19.26  1593  alsyl  1615  vtoclgft  2906  euind  3024  reuind  3040  sbeqalb  3099