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Mirrors > Home > MPE Home > Th. List > ss2ralv | Structured version Visualization version GIF version |
Description: Two quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023.) |
Ref | Expression |
---|---|
ss2ralv | ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssralv 3953 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐴 𝜑)) | |
2 | 1 | ralimdv 3091 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝜑)) |
3 | ssralv 3953 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)) | |
4 | 2, 3 | syld 47 | 1 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wral 3051 ⊆ wss 3853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-v 3400 df-in 3860 df-ss 3870 |
This theorem is referenced by: poss 5455 soss 5473 dffi3 9025 isercolllem1 15193 cfilres 24147 lgsdchr 26190 dffltz 40115 |
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