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Mirrors > Home > ILE Home > Th. List > eqsstrri | GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.) |
Ref | Expression |
---|---|
eqsstr3.1 | ⊢ 𝐵 = 𝐴 |
eqsstr3.2 | ⊢ 𝐵 ⊆ 𝐶 |
Ref | Expression |
---|---|
eqsstrri | ⊢ 𝐴 ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstr3.1 | . . 3 ⊢ 𝐵 = 𝐴 | |
2 | 1 | eqcomi 2144 | . 2 ⊢ 𝐴 = 𝐵 |
3 | eqsstr3.2 | . 2 ⊢ 𝐵 ⊆ 𝐶 | |
4 | 2, 3 | eqsstri 3134 | 1 ⊢ 𝐴 ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ⊆ wss 3076 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-in 3082 df-ss 3089 |
This theorem is referenced by: inss2 3302 dmv 4763 resasplitss 5310 ofrfval 5998 ofvalg 5999 ofrval 6000 off 6002 ofres 6004 ofco 6008 dftpos4 6168 smores2 6199 caseinj 6982 djuinj 6999 bcm1k 10538 bcpasc 10544 |
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