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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 1181 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜑) |
3 | 2 | ralrimiva 2505 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜑) |
4 | 3 | rgen2 2518 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 |
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: reg3exmidlemwe 4493 ltsopr 7404 ltsosr 7572 ltso 7842 xrltso 9582 addcncntoplem 12720 |
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