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Mirrors > Home > NFE Home > Th. List > 2ralbidva | GIF version |
Description: Formula-building rule for restricted universal quantifiers (deduction rule). (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 1619 | . 2 ⊢ Ⅎxφ | |
2 | nfv 1619 | . 2 ⊢ Ⅎyφ | |
3 | 2ralbidva.1 | . 2 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ)) | |
4 | 1, 2, 3 | 2ralbida 2653 | 1 ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∈ wcel 1710 ∀wral 2614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-ral 2619 |
This theorem is referenced by: (None) |
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