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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 3131 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
3 | sseq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | sseq2d 3132 | . 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 1332 ⊆ wss 3076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-in 3082 df-ss 3089 |
This theorem is referenced by: 3sstr3d 3146 3sstr4d 3147 ssdifeq0 3450 relcnvtr 5066 rdgisucinc 6290 oawordriexmid 6374 nnaword 6415 nnawordi 6419 sbthlem2 6854 isbth 6863 infnninf 7030 nnnninf 7031 ennnfonelemkh 11961 ennnfonelemrnh 11965 isstruct2im 12008 isstruct2r 12009 ressid2 12057 ressval2 12058 basis1 12253 baspartn 12256 eltg 12260 metss 12702 0nninf 13372 nninff 13373 nnsf 13374 peano4nninf 13375 nninfalllemn 13377 nninfalllem1 13378 nninfself 13384 |
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