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Mirrors > Home > MPE Home > Th. List > eqeltrd | Structured version Visualization version GIF version |
Description: Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
eqeltrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqeltrd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
Ref | Expression |
---|---|
eqeltrd | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeltrd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
2 | eqeltrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | eleq1d 2822 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) |
4 | 1, 3 | mpbird 260 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
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