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| Mirrors > Home > MPE Home > Th. List > rabeqf | Structured version Visualization version GIF version | ||
| Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) |
| Ref | Expression |
|---|---|
| rabeqf.1 | ⊢ Ⅎ𝑥𝐴 |
| rabeqf.2 | ⊢ Ⅎ𝑥𝐵 |
| Ref | Expression |
|---|---|
| rabeqf | ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabeqf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 2 | rabeqf.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 3 | 1, 2 | nfeq 2919 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
| 4 | eleq2 2830 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
| 5 | 4 | anbi1d 631 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
| 6 | 3, 5 | abbid 2810 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
| 7 | df-rab 3437 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
| 8 | df-rab 3437 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
| 9 | 6, 7, 8 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 {crab 3436 |
| 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-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 |
| This theorem is referenced by: fpwrelmapffs 32745 rabeq12f 38164 issmfdf 46752 smfpimltmpt 46761 smfpimltxrmptf 46773 smfpimgtmpt 46796 smfpimgtxrmptf 46799 smfsupmpt 46830 |
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