Theorem rr19.28v 2982

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 3646 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 443	. . . . . 6 ⊢ ((φ ∧ ψ) → φ)
2	1	ralimi 2690	. . . . 5 ⊢ (∀y ∈ A (φ ∧ ψ) → ∀y ∈ A φ)
3		biidd 228	. . . . . 6 ⊢ (y = x → (φ ↔ φ))
4	3	rspcv 2952	. . . . 5 ⊢ (x ∈ A → (∀y ∈ A φ → φ))
5	2, 4	syl5 28	. . . 4 ⊢ (x ∈ A → (∀y ∈ A (φ ∧ ψ) → φ))
6		simpr 447	. . . . . 6 ⊢ ((φ ∧ ψ) → ψ)
7	6	ralimi 2690	. . . . 5 ⊢ (∀y ∈ A (φ ∧ ψ) → ∀y ∈ A ψ)
8	7	a1i 10	. . . 4 ⊢ (x ∈ A → (∀y ∈ A (φ ∧ ψ) → ∀y ∈ A ψ))
9	5, 8	jcad 519	. . 3 ⊢ (x ∈ A → (∀y ∈ A (φ ∧ ψ) → (φ ∧ ∀y ∈ A ψ)))
10	9	ralimia 2688	. 2 ⊢ (∀x ∈ A ∀y ∈ A (φ ∧ ψ) → ∀x ∈ A (φ ∧ ∀y ∈ A ψ))
11		r19.28av 2754	. . 3 ⊢ ((φ ∧ ∀y ∈ A ψ) → ∀y ∈ A (φ ∧ ψ))
12	11	ralimi 2690	. 2 ⊢ (∀x ∈ A (φ ∧ ∀y ∈ A ψ) → ∀x ∈ A ∀y ∈ A (φ ∧ ψ))
13	10, 12	impbii 180	1 ⊢ (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ ∀x ∈ A (φ ∧ ∀y ∈ A ψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862

This theorem is used by: (None)