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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 1229 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜓) |
3 | 2 | ralrimiva 2550 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜓) |
4 | 3 | ralrimiva 2550 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓) |
5 | 4 | ralrimiva 2550 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 ∈ wcel 2148 ∀wral 2455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-4 1510 ax-17 1526 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-nf 1461 df-ral 2460 |
This theorem is referenced by: ispod 4300 swopolem 4301 ordwe 4571 wessep 4573 isopolem 5816 caovassg 6026 caovcang 6029 caovordig 6033 caovordg 6035 caovdig 6042 caovdirg 6045 caoftrn 6101 isringd 13033 |
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