Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3123 | . 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 1331 ⊆ wss 3066 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-in 3072 df-ss 3079 |
This theorem is referenced by: rdgss 6273 sucinc2 6335 oawordi 6358 nnnninf 7016 fzoss1 9941 fzoss2 9942 clsss 12276 ntrss 12277 sslm 12405 txss12 12424 metss2lem 12655 xmettxlem 12667 xmettx 12668 nnsf 13188 nninfself 13198 |
Copyright terms: Public domain | W3C validator |