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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 1257 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜑) |
3 | 2 | ralrimiva 2949 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜑) |
4 | 3 | rgen2 2958 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ∀wral 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 |
This theorem depends on definitions: df-bi 196 df-an 385 df-3an 1033 df-ral 2901 |
This theorem is referenced by: isposi 16779 addcnlem 22475 isgrpoi 26736 lnocoi 26996 0lnfn 28228 lnopcoi 28246 xrge0omnd 29042 reofld 29171 poseq 30994 2zrngasgrp 41730 2zrngmsgrp 41737 2zrngALT 41738 |
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