Description: Alternate proof of eleq2d , shorter at the expense of requiring ax-12 . (Contributed by NM, 27-Dec-1993) (Revised by Wolf Lammen, 20-Nov-2019) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | eleq1d.1 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |
| Assertion | eleq2dALT | ⊢ ( 𝜑 → ( 𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq1d.1 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |
| 2 | dfcleq | ⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) | |
| 3 | 1 2 | sylib | ⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
| 4 | 3 | 19.21bi | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) |
| 5 | 4 | anbi2d | ⊢ ( 𝜑 → ( ( 𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵 ) ) ) |
| 6 | 5 | exbidv | ⊢ ( 𝜑 → ( ∃ 𝑥 ( 𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴 ) ↔ ∃ 𝑥 ( 𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵 ) ) ) |
| 7 | dfclel | ⊢ ( 𝐶 ∈ 𝐴 ↔ ∃ 𝑥 ( 𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴 ) ) | |
| 8 | dfclel | ⊢ ( 𝐶 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵 ) ) | |
| 9 | 6 7 8 | 3bitr4g | ⊢ ( 𝜑 → ( 𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵 ) ) |