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Mirrors > Home > MPE Home > Th. List > 2ralbida | Structured version Visualization version 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 1830 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
4 | 2, 3 | nfan 1816 | . . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴) |
5 | 2ralbida.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) | |
6 | 5 | anassrs 678 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 ↔ 𝜒)) |
7 | 4, 6 | ralbida 2965 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)) |
8 | 1, 7 | ralbida 2965 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-ral 2901 |
This theorem is referenced by: (None) |
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