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Mirrors > Home > ILE Home > Th. List > rgen3 | GIF version |
Description: Generalization rule for restricted quantification. (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 1138 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜑) |
3 | 2 | ralrimiva 2434 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜑) |
4 | 3 | rgen2 2447 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 919 ∈ wcel 1433 ∀wral 2348 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-4 1440 ax-17 1459 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-nf 1390 df-ral 2353 |
This theorem is referenced by: reg3exmidlemwe 4321 ltsopr 6786 ltsosr 6941 ltso 7189 xrltso 8871 |
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