Theorem rr19.3v 3345

Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4064 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 252	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
2	1	rspcv 3305	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝜑 → 𝜑))
3	2	ralimia 2950	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
4		ax-1 6	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝜑))
5	4	ralrimiv 2965	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝜑)
6	5	ralimi 2952	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)
7	3, 6	impbii 199	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 196   ∈ wcel 1990  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by:  ispos2  16948