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Mirrors > Home > MPE Home > Th. List > rr19.3v | Structured version Visualization version GIF version |
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4440 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.) |
Ref | Expression |
---|---|
rr19.3v | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 263 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜑)) | |
2 | 1 | rspcv 3615 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝜑 → 𝜑)) |
3 | 2 | ralimia 3155 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
4 | ax-1 6 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝜑)) | |
5 | 4 | ralrimiv 3178 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝜑) |
6 | 5 | ralimi 3157 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) |
7 | 3, 6 | impbii 210 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∈ wcel 2105 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-clel 2890 df-ral 3140 |
This theorem is referenced by: ispos2 17546 |
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