Description: Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ovanraleqv.1 | ⊢ ( 𝐵 = 𝑋 → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | ovanraleqv | ⊢ ( 𝐵 = 𝑋 → ( ∀ 𝑥 ∈ 𝑉 ( 𝜑 ∧ ( 𝐴 · 𝐵 ) = 𝐶 ) ↔ ∀ 𝑥 ∈ 𝑉 ( 𝜓 ∧ ( 𝐴 · 𝑋 ) = 𝐶 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ovanraleqv.1 | ⊢ ( 𝐵 = 𝑋 → ( 𝜑 ↔ 𝜓 ) ) | |
| 2 | oveq2 | ⊢ ( 𝐵 = 𝑋 → ( 𝐴 · 𝐵 ) = ( 𝐴 · 𝑋 ) ) | |
| 3 | 2 | eqeq1d | ⊢ ( 𝐵 = 𝑋 → ( ( 𝐴 · 𝐵 ) = 𝐶 ↔ ( 𝐴 · 𝑋 ) = 𝐶 ) ) |
| 4 | 1 3 | anbi12d | ⊢ ( 𝐵 = 𝑋 → ( ( 𝜑 ∧ ( 𝐴 · 𝐵 ) = 𝐶 ) ↔ ( 𝜓 ∧ ( 𝐴 · 𝑋 ) = 𝐶 ) ) ) |
| 5 | 4 | ralbidv | ⊢ ( 𝐵 = 𝑋 → ( ∀ 𝑥 ∈ 𝑉 ( 𝜑 ∧ ( 𝐴 · 𝐵 ) = 𝐶 ) ↔ ∀ 𝑥 ∈ 𝑉 ( 𝜓 ∧ ( 𝐴 · 𝑋 ) = 𝐶 ) ) ) |