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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 2823 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)) |
3 | eleq12d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | eleq1d 2822 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷)) |
5 | 2, 4 | bitrd 282 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
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