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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 3225 | . . 3 ⊢ (𝑅 = 𝑆 → 𝑆 ⊆ 𝑅) | |
2 | sess1 4355 | . . 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 3224 | . . 3 ⊢ (𝑅 = 𝑆 → 𝑅 ⊆ 𝑆) | |
5 | sess1 4355 | . . 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 129 | 1 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ⊆ wss 3144 Se wse 4347 |
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 ax-sep 4136 |
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-ral 2473 df-rab 2477 df-v 2754 df-in 3150 df-ss 3157 df-br 4019 df-se 4351 |
This theorem is referenced by: (None) |
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