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| Mirrors > Home > MPE Home > Th. List > opabbii | Structured version Visualization version GIF version | ||
| Description: Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.) |
| Ref | Expression |
|---|---|
| opabbii.1 | ⊢ (𝜑 ↔ 𝜓) |
| Ref | Expression |
|---|---|
| opabbii | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ 𝑧 = 𝑧 | |
| 2 | opabbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓)) |
| 4 | 3 | opabbidv 5209 | . 2 ⊢ (𝑧 = 𝑧 → {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
| 5 | 1, 4 | ax-mp 5 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} |
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