![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > seeq1 | GIF version |
Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
seeq1 | ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 3116 | . . 3 ⊢ (𝑅 = 𝑆 → 𝑆 ⊆ 𝑅) | |
2 | sess1 4217 | . . 3 ⊢ (𝑆 ⊆ 𝑅 → (𝑅 Se 𝐴 → 𝑆 Se 𝐴)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 → 𝑆 Se 𝐴)) |
4 | eqimss 3115 | . . 3 ⊢ (𝑅 = 𝑆 → 𝑅 ⊆ 𝑆) | |
5 | sess1 4217 | . . 3 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝑅 = 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) |
7 | 3, 6 | impbid 128 | 1 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1312 ⊆ wss 3035 Se wse 4209 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rab 2397 df-v 2657 df-in 3041 df-ss 3048 df-br 3894 df-se 4213 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |