Theorem alanimi 1470

Description: Variant of al2imi 1469 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 115	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1469	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	3	imp 124	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460

This theorem is referenced by:  19.26  1492  alsyl  1646  vtoclgft  2802  euind  2939  reuind  2957  sbeqalb  3034  bm1.3ii  4139  trin2  5038  bdbm1.3ii  15101