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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 3126 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
3 | sseq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | sseq2d 3127 | . 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 1331 ⊆ wss 3071 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: 3sstr3d 3141 3sstr4d 3142 ssdifeq0 3445 relcnvtr 5058 rdgisucinc 6282 oawordriexmid 6366 nnaword 6407 nnawordi 6411 sbthlem2 6846 isbth 6855 infnninf 7022 nnnninf 7023 ennnfonelemkh 11925 ennnfonelemrnh 11929 isstruct2im 11969 isstruct2r 11970 ressid2 12018 ressval2 12019 basis1 12214 baspartn 12217 eltg 12221 metss 12663 0nninf 13197 nninff 13198 nnsf 13199 peano4nninf 13200 nninfalllemn 13202 nninfalllem1 13203 nninfself 13209 |
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