Theorem rr19.3v 2981

Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 3644 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)

Assertion

Ref	Expression
rr19.3v	⊢ (∀x ∈ A ∀y ∈ A φ ↔ ∀x ∈ A φ)

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

Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem rr19.3v

Step	Hyp	Ref	Expression
1		biidd 228	. . . 4 ⊢ (y = x → (φ ↔ φ))
2	1	rspcv 2952	. . 3 ⊢ (x ∈ A → (∀y ∈ A φ → φ))
3	2	ralimia 2688	. 2 ⊢ (∀x ∈ A ∀y ∈ A φ → ∀x ∈ A φ)
4		ax-1 6	. . . 4 ⊢ (φ → (y ∈ A → φ))
5	4	ralrimiv 2697	. . 3 ⊢ (φ → ∀y ∈ A φ)
6	5	ralimi 2690	. 2 ⊢ (∀x ∈ A φ → ∀x ∈ A ∀y ∈ A φ)
7	3, 6	impbii 180	1 ⊢ (∀x ∈ A ∀y ∈ A φ ↔ ∀x ∈ A φ)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2615

This theorem was proved from 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 theorem 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 referenced by: (None)