Description: All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020) Avoid ax-8 . (Revised by Wolf Lammen, 9-Mar-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | raleleq | ⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfcleq | ⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) | |
| 2 | biimp | ⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) | |
| 3 | 2 | alimi | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
| 4 | 1 3 | sylbi | ⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
| 5 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) | |
| 6 | 4 5 | sylibr | ⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ) |