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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 1508 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1508 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | 2ralbidva.1 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) | |
4 | 1, 2, 3 | 2ralbida 2478 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2128 ∀wral 2435 |
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 1427 ax-gen 1429 ax-4 1490 ax-17 1506 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-ral 2440 |
This theorem is referenced by: soinxp 4658 isotr 5768 fnmpoovd 6164 ismet2 12824 txmetcn 12989 |
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