Theorem rr19.28v 3626

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4466 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)

Assertion

Ref	Expression
rr19.28v	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥, 𝑦)   𝐴(𝑥)

Proof of Theorem rr19.28v

Step	Hyp	Ref	Expression
1		simpl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	ralimi 3101	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐴 𝜑)
3		biidd 265	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
4	3	rspcv 3576	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝜑 → 𝜑))
5	2, 4	syl5 35	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → 𝜑))
6		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
7	6	ralimi 3101	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐴 𝜓)
8	5, 7	jca2 522	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
9	8	ralimia 3098	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
10		r19.28v 3195	. . 3 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓) → ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
11	10	ralimi 3101	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
12	9, 11	impbii 212	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ∈ wcel 2142  ∀wral 3078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079

This theorem is used by: (None)