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Mirrors > Home > ILE Home > Th. List > 2ralbida | GIF version |
Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.) |
Ref | Expression |
---|---|
2ralbida.1 | ⊢ Ⅎxφ |
2ralbida.2 | ⊢ Ⅎyφ |
2ralbida.3 | ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ)) |
Ref | Expression |
---|---|
2ralbida | ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ralbida.1 | . 2 ⊢ Ⅎxφ | |
2 | 2ralbida.2 | . . . 4 ⊢ Ⅎyφ | |
3 | nfv 1418 | . . . 4 ⊢ Ⅎy x ∈ A | |
4 | 2, 3 | nfan 1454 | . . 3 ⊢ Ⅎy(φ ∧ x ∈ A) |
5 | 2ralbida.3 | . . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ)) | |
6 | 5 | anassrs 380 | . . 3 ⊢ (((φ ∧ x ∈ A) ∧ y ∈ B) → (ψ ↔ χ)) |
7 | 4, 6 | ralbida 2314 | . 2 ⊢ ((φ ∧ x ∈ A) → (∀y ∈ B ψ ↔ ∀y ∈ B χ)) |
8 | 1, 7 | ralbida 2314 | 1 ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 Ⅎwnf 1346 ∈ wcel 1390 ∀wral 2300 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-4 1397 ax-17 1416 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-ral 2305 |
This theorem is referenced by: 2ralbidva 2340 |
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