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| Mirrors > Home > MPE Home > Th. List > ralbidv | Structured version Visualization version GIF version | ||
| Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.) |
| Ref | Expression |
|---|---|
| ralbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| ralbidv | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralbidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
| 3 | 2 | ralbidva 3176 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
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