Description: Alternate proof of eleq2w2 and special instance of eleq2 . (Contributed by BJ, 22-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | eleq2w2ALT | ⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfcleq | ⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) ) | |
| 2 | 1 | biimpi | ⊢ ( 𝐴 = 𝐵 → ∀ 𝑦 ( 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) ) |
| 3 | eleq1w | ⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴 ) ) | |
| 4 | eleq1w | ⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵 ) ) | |
| 5 | 3 4 | bibi12d | ⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) ) |
| 6 | 5 | spvv | ⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
| 7 | 2 6 | syl | ⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |