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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 3188 | . 2 ⊢ (𝜑 → (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵)) |
5 | 1, 4 | mpbird 167 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ⊆ wss 3131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3137 df-ss 3144 |
This theorem is referenced by: rdgss 6387 sucinc2 6450 oawordi 6473 nnnninf 7127 fzoss1 10174 fzoss2 10175 lspss 13524 clsss 13806 ntrss 13807 sslm 13935 txss12 13954 metss2lem 14185 xmettxlem 14197 xmettx 14198 nnsf 14943 nninfself 14951 |
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