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Mirrors > Home > MPE Home > Th. List > rgen3 | Structured version Visualization version GIF version |
Description: Generalization rule for restricted quantification, with three quantifiers. (Contributed by NM, 12-Jan-2008.) |
Ref | Expression |
---|---|
rgen3.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜑) |
Ref | Expression |
---|---|
rgen3 | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rgen3.1 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜑) | |
2 | 1 | 3expa 1116 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜑) |
3 | 2 | ralrimiva 3142 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜑) |
4 | 3 | rgen2 3193 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2099 ∀wral 3057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1087 df-ral 3058 |
This theorem is referenced by: poseq 8158 isposi 18310 efmndsgrp 18832 smndex1sgrp 18854 addcnlem 24774 addscutlem 27888 isgrpoi 30302 lnocoi 30561 0lnfn 31789 lnopcoi 31807 xrge0omnd 32786 reofld 33051 2zrngasgrp 47299 2zrngmsgrp 47306 2zrngALT 47307 |
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