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| Mirrors > Home > MPE Home > Th. List > abbidv | Structured version Visualization version GIF version | ||
| Description: Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.) Avoid ax-12 2177, based on an idea of Steven Nguyen. (Revised by Wolf Lammen, 6-May-2023.) |
| Ref | Expression |
|---|---|
| abbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| abbidv | ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | abbidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | alrimiv 1927 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
| 3 | abbi 2807 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | |
| 4 | 2, 3 | syl 17 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
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