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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 2913 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | eleq2 2818 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 4 | anbi1d 630 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
6 | 3, 5 | abbid 2799 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
7 | df-rab 3430 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
8 | df-rab 3430 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
9 | 6, 7, 8 | 3eqtr4g 2793 | 1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {cab 2705 Ⅎwnfc 2879 {crab 3429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-rab 3430 |
This theorem is referenced by: fpwrelmapffs 32529 rabeq12f 37630 issmfdf 46125 smfpimltmpt 46134 smfpimltxrmptf 46146 smfpimgtmpt 46169 smfpimgtxrmptf 46172 smfsupmpt 46203 smfinfmpt 46207 |
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