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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 3076 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
4 | 1, 3 | mpbird 166 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ⊆ wss 3021 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-11 1452 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-in 3027 df-ss 3034 |
This theorem is referenced by: eqsstr3d 3084 syl6eqss 3099 tfisi 4439 funresdfunsnss 5555 suppssof1 5930 phplem4dom 6685 cardonle 6954 exmidfodomrlemim 6966 frecuzrdgtclt 10035 ennnfonelemkh 11717 ennnfonelemf1 11723 strfvssn 11763 setscom 11781 tgrest 12120 resttopon 12122 rest0 12130 lmtopcnp 12200 metequiv2 12424 ellimc3ap 12511 dvfvalap 12523 nnsf 12783 |
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