Theorem rr19.3v 3661

Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4444 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 264	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
2	1	rspcv 3618	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝜑 → 𝜑))
3	2	ralimia 3158	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
4		ax-1 6	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝜑))
5	4	ralrimiv 3181	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝜑)
6	5	ralimi 3160	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)
7	3, 6	impbii 211	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 208   ∈ wcel 2114  ∀wral 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2814  df-clel 2893  df-ral 3143

This theorem is referenced by:  ispos2  17558