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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 2918 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | eleq2 2826 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 4 | anbi1d 630 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
6 | 3, 5 | abbid 2807 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
7 | df-rab 3406 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
8 | df-rab 3406 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
9 | 6, 7, 8 | 3eqtr4g 2801 | 1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2713 Ⅎwnfc 2885 {crab 3405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-rab 3406 |
This theorem is referenced by: fpwrelmapffs 31534 rabeq12f 36583 issmfdf 44910 smfpimltmpt 44919 smfpimltxrmptf 44931 smfpimgtmpt 44954 smfpimgtxrmptf 44957 smfsupmpt 44988 smfinfmpt 44992 |
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