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Mirrors > Home > ILE Home > Th. List > ralrimivvva | GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ralrimivvva.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) → 𝜓) |
Ref | Expression |
---|---|
ralrimivvva | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrimivvva.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) → 𝜓) | |
2 | 1 | 3anassrs 1207 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜓) |
3 | 2 | ralrimiva 2505 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜓) |
4 | 3 | ralrimiva 2505 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓) |
5 | 4 | ralrimiva 2505 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 ∈ wcel 1480 ∀wral 2416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-4 1487 ax-17 1506 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-nf 1437 df-ral 2421 |
This theorem is referenced by: ispod 4226 swopolem 4227 ordwe 4490 wessep 4492 isopolem 5723 caovassg 5929 caovcang 5932 caovordig 5936 caovordg 5938 caovdig 5945 caovdirg 5948 caoftrn 6007 |
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