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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 4402 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 265 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜑)) | |
2 | 1 | rspcv 3566 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝜑 → 𝜑)) |
3 | 2 | ralimia 3126 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
4 | ax-1 6 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝜑)) | |
5 | 4 | ralrimiv 3148 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝜑) |
6 | 5 | ralimi 3128 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) |
7 | 3, 6 | impbii 212 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2111 ∀wral 3106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 df-clel 2870 df-ral 3111 |
This theorem is referenced by: ispos2 17550 |
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