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| Mirrors > Home > MPE Home > Th. List > eleq2s | Structured version Visualization version GIF version | ||
| Description: Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
| Ref | Expression |
|---|---|
| eleq2s.1 | ⊢ (𝐴 ∈ 𝐵 → 𝜑) |
| eleq2s.2 | ⊢ 𝐶 = 𝐵 |
| Ref | Expression |
|---|---|
| eleq2s | ⊢ (𝐴 ∈ 𝐶 → 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq2s.2 | . . 3 ⊢ 𝐶 = 𝐵 | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐵) |
| 3 | eleq2s.1 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝜑) | |
| 4 | 2, 3 | sylbi 217 | 1 ⊢ (𝐴 ∈ 𝐶 → 𝜑) |
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