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Mirrors > Home > ILE Home > Th. List > ralbida | GIF version |
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.) |
Ref | Expression |
---|---|
ralbida.1 | ⊢ Ⅎ𝑥𝜑 |
ralbida.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralbida | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbida.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ralbida.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | pm5.74da 443 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜒))) |
4 | 1, 3 | albid 1615 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))) |
5 | df-ral 2460 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
6 | df-ral 2460 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)) | |
7 | 4, 5, 6 | 3bitr4g 223 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1351 Ⅎwnf 1460 ∈ 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 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-ral 2460 |
This theorem is referenced by: ralbidva 2473 ralbid 2475 2ralbida 2498 ralbi 2609 ismkvnex 7147 caucvgsrlemgt1 7782 iswomninnlem 14446 |
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