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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 2966 | . 2 ⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐶)) |
3 | sseq12d.2 | . . 3 ⊢ (φ → 𝐶 = 𝐷) | |
4 | 3 | sseq2d 2967 | . 2 ⊢ (φ → (B ⊆ 𝐶 ↔ B ⊆ 𝐷)) |
5 | 2, 4 | bitrd 177 | 1 ⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐷)) |
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: 3sstr3d 2981 3sstr4d 2982 ssdifeq0 3299 relcnvtr 4783 rdgisucinc 5912 nnaword 6020 nnawordi 6024 |
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