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Mirrors > Home > ILE Home > Th. List > 2ralbidva | GIF version |
Description: Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.) |
Ref | Expression |
---|---|
2ralbidva.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
2ralbidva | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1528 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1528 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | 2ralbidva.1 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) | |
4 | 1, 2, 3 | 2ralbida 2498 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ 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-nf 1461 df-ral 2460 |
This theorem is referenced by: soinxp 4692 isotr 5810 fnmpoovd 6209 mndpropd 12720 mhmpropd 12734 cmnpropd 12912 ringpropd 13030 ismet2 13487 txmetcn 13652 |
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