Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > raleqbidv | Structured version Visualization version GIF version |
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
Ref | Expression |
---|---|
raleqbidv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
raleqbidv.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
raleqbidv | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqbidv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | raleqdv 3121 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
3 | raleqbidv.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 3 | ralbidv 2969 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
5 | 2, 4 | bitrd 267 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Copyright terms: Public domain | W3C validator |