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| Mirrors > Home > MPE Home > Th. List > eleq12d | Structured version Visualization version GIF version | ||
| Description: Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
| Ref | Expression |
|---|---|
| eleq12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| eleq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| eleq12d | ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 2 | 1 | eleq2d 2827 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)) |
| 3 | eleq12d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 4 | 3 | eleq1d 2826 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷)) |
| 5 | 2, 4 | bitrd 279 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
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