![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rabeqif | GIF version |
Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2742. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
rabeqf.1 | ⊢ Ⅎ𝑥𝐴 |
rabeqf.2 | ⊢ Ⅎ𝑥𝐵 |
rabeqif.3 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
rabeqif | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqif.3 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | rabeqf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | rabeqf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | rabeqf 2742 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 Ⅎwnfc 2319 {crab 2472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rab 2477 |
This theorem is referenced by: rabeqi 2745 |
Copyright terms: Public domain | W3C validator |