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| Mirrors > Home > MPE Home > Th. List > abbid | Structured version Visualization version GIF version | ||
| Description: Equivalent wff's yield equal class abstractions (deduction form, with nonfreeness hypothesis). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) Avoid ax-10 2141 and ax-11 2157. (Revised by Wolf Lammen, 6-May-2023.) |
| Ref | Expression |
|---|---|
| abbid.1 | ⊢ Ⅎ𝑥𝜑 |
| abbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| abbid | ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | abbid.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | abbid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 3 | 1, 2 | alrimi 2213 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
| 4 | abbi 2807 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | |
| 5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 {cab 2714 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 |
| This theorem is referenced by: rabbida4 3462 rabeqf 3472 sbcbid 3844 sbceqbidf 32506 opabdm 32623 opabrn 32624 fpwrelmap 32744 sticksstones16 42163 rabbida2 45137 rabbida3 45140 |
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