![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sseq1d | GIF version |
Description: An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
sseq1d.1 | ⊢ (φ → A = B) |
Ref | Expression |
---|---|
sseq1d | ⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1d.1 | . 2 ⊢ (φ → A = B) | |
2 | sseq1 2960 | . 2 ⊢ (A = B → (A ⊆ 𝐶 ↔ B ⊆ 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 |
This theorem is referenced by: sseq12d 2968 eqsstrd 2973 snssg 3491 ssiun2s 3692 treq 3851 onsucsssucexmid 4212 funimass1 4919 feq1 4973 sbcfg 4988 fvmptssdm 5198 fvimacnvi 5224 nnsucsssuc 6010 ereq1 6049 |
Copyright terms: Public domain | W3C validator |