Metamath Proof Explorer
Description: Version of raleqbidv with additional disjoint variable conditions, not
requiring ax-8 nor df-clel . (Contributed by BJ, 22-Sep-2024)
|
|
Ref |
Expression |
|
Hypotheses |
raleqbidvv.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
|
|
raleqbidvv.2 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
|
Assertion |
raleqbidvv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
raleqbidvv.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
2 |
|
raleqbidvv.2 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
3 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) |
4 |
1 3
|
raleqbidva |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) ) |