Theorem rr19.3v 2965

Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (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 172	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
2	1	rspcv 2925	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝜑 → 𝜑))
3	2	ralimia 2611	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
4		ax-1 6	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝜑))
5	4	ralrimiv 2622	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝜑)
6	5	ralimi 2613	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)
7	3, 6	impbii 126	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 105   ∈ wcel 2209  ∀wral 2528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823

This theorem is referenced by: (None)