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Mirrors > Home > ILE Home > Th. List > eqsstrd | GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
eqsstrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqsstrd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
eqsstrd | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstrd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
2 | eqsstrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | sseq1d 2972 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
4 | 1, 3 | mpbird 156 | 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: eqsstr3d 2980 syl6eqss 2995 tfisi 4310 suppssof1 5728 phplem4dom 6324 cardonle 6367 |
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