![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sseq12d | GIF version |
Description: An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.) |
Ref | Expression |
---|---|
sseq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
sseq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
sseq12d | ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | sseq1d 3076 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
3 | sseq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | sseq2d 3077 | . 2 ⊢ (𝜑 → (𝐵 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) |
5 | 2, 4 | bitrd 187 | 1 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1299 ⊆ wss 3021 |
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-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-11 1452 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-in 3027 df-ss 3034 |
This theorem is referenced by: 3sstr3d 3091 3sstr4d 3092 ssdifeq0 3392 relcnvtr 4994 rdgisucinc 6212 oawordriexmid 6296 nnaword 6337 nnawordi 6341 sbthlem2 6774 isbth 6783 infnninf 6934 nnnninf 6935 ennnfonelemkh 11717 ennnfonelemrnh 11721 isstruct2im 11751 isstruct2r 11752 ressid2 11800 ressval2 11801 basis1 11996 baspartn 11999 eltg 12003 metss 12422 0nninf 12781 nninff 12782 nnsf 12783 peano4nninf 12784 nninfalllemn 12786 nninfalllem1 12787 nninfself 12793 |
Copyright terms: Public domain | W3C validator |