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Mirrors > Home > ILE Home > Th. List > 3sstr4d | GIF version |
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
Ref | Expression |
---|---|
3sstr4d.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
3sstr4d.2 | ⊢ (𝜑 → 𝐶 = 𝐴) |
3sstr4d.3 | ⊢ (𝜑 → 𝐷 = 𝐵) |
Ref | Expression |
---|---|
3sstr4d | ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3sstr4d.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 3sstr4d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐴) | |
3 | 3sstr4d.3 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐵) | |
4 | 2, 3 | sseq12d 3094 | . 2 ⊢ (𝜑 → (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵)) |
5 | 1, 4 | mpbird 166 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 ⊆ wss 3037 |
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 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-11 1467 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-in 3043 df-ss 3050 |
This theorem is referenced by: rdgss 6234 sucinc2 6296 oawordi 6319 nnnninf 6973 fzoss1 9841 fzoss2 9842 clsss 12130 ntrss 12131 sslm 12258 txss12 12277 metss2lem 12486 xmettxlem 12498 xmettx 12499 nnsf 12891 nninfself 12901 |
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