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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 2144 and ax-11 2160. (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 2212 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | abbi1 2887 | . 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 208 ∀wal 1534 = wceq 1536 Ⅎwnf 1783 {cab 2802 |
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 1969 ax-7 2014 ax-9 2123 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 |
This theorem is referenced by: rabeqf 3484 rabeqi 3485 sbcbid 3829 sbceqbidf 30253 opabdm 30365 opabrn 30366 fpwrelmap 30472 bj-rabbida2 34241 rabbida2 41405 rabbida3 41408 |
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