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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssralv2 | Structured version Visualization version GIF version |
Description: Quantification restricted to a subclass for two quantifiers. ssralv 3981 for two quantifiers. The proof of ssralv2 41237 was automatically generated by minimizing the automatically translated proof of ssralv2VD 41572. The automatic translation is by the tools program translate_without_overwriting.cmd. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ssralv2 | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . 2 ⊢ Ⅎ𝑥(𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) | |
2 | nfra1 3183 | . 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 | |
3 | ssralv 3981 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑)) | |
4 | 3 | adantr 484 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑)) |
5 | df-ral 3111 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑)) | |
6 | 4, 5 | syl6ib 254 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑))) |
7 | sp 2180 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑) → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑)) | |
8 | 6, 7 | syl6 35 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷 𝜑))) |
9 | ssralv 3981 | . . . 4 ⊢ (𝐶 ⊆ 𝐷 → (∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐶 𝜑)) | |
10 | 9 | adantl 485 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐶 𝜑)) |
11 | 8, 10 | syl6d 75 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶 𝜑))) |
12 | 1, 2, 11 | ralrimd 3182 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 |
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-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 |
This theorem is referenced by: ordelordALT 41243 ordelordALTVD 41573 |
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