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Mirrors > Home > ILE Home > Th. List > syl6eqss | GIF version |
Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
syl6eqss.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
syl6eqss.2 | ⊢ 𝐵 ⊆ 𝐶 |
Ref | Expression |
---|---|
syl6eqss | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6eqss.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | syl6eqss.2 | . . 3 ⊢ 𝐵 ⊆ 𝐶 | |
3 | 2 | a1i 9 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
4 | 1, 3 | eqsstrd 2979 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ⊆ wss 2917 |
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 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 |
This theorem is referenced by: syl6eqssr 2996 resasplitss 5069 fimacnv 5296 bj-nntrans 10076 |
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