Description: Equality deduction for restricted universal quantifier. See raleqbidv for a version based on fewer axioms. (Contributed by Thierry Arnoux, 8-Mar-2017)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | raleqbid.0 | ⊢ Ⅎ 𝑥 𝜑 | |
| raleqbid.1 | ⊢ Ⅎ 𝑥 𝐴 | ||
| raleqbid.2 | ⊢ Ⅎ 𝑥 𝐵 | ||
| raleqbid.3 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | ||
| raleqbid.4 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | ||
| Assertion | raleqbid | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | raleqbid.0 | ⊢ Ⅎ 𝑥 𝜑 | |
| 2 | raleqbid.1 | ⊢ Ⅎ 𝑥 𝐴 | |
| 3 | raleqbid.2 | ⊢ Ⅎ 𝑥 𝐵 | |
| 4 | raleqbid.3 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |
| 5 | raleqbid.4 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
| 6 | 2 3 | raleqf | ⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) ) |
| 7 | 4 6 | syl | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) ) |
| 8 | 1 5 | ralbid | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) ) |
| 9 | 7 8 | bitrd | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) ) |