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Mirrors > Home > MPE Home > Th. List > eqsstr3i | Structured version Visualization version 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 |
---|---|
eqsstr3i | ⊢ 𝐴 ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstr3.1 | . . 3 ⊢ 𝐵 = 𝐴 | |
2 | 1 | eqcomi 2780 | . 2 ⊢ 𝐴 = 𝐵 |
3 | eqsstr3.2 | . 2 ⊢ 𝐵 ⊆ 𝐶 | |
4 | 2, 3 | eqsstri 3784 | 1 ⊢ 𝐴 ⊆ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ⊆ wss 3723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-in 3730 df-ss 3737 |
This theorem is referenced by: inss2 3982 dmv 5478 ofrfval 7056 ofval 7057 ofrval 7058 off 7063 ofres 7064 ofco 7068 dftpos4 7527 smores2 7608 onwf 8861 r0weon 9039 cda1dif 9204 unctb 9233 infmap2 9246 itunitc 9449 axcclem 9485 dfnn3 11240 cotr2 13926 ressbasss 16139 strlemor1OLD 16177 prdsle 16330 prdsless 16331 dprd2da 18649 opsrle 19690 indiscld 21116 leordtval2 21237 fiuncmp 21428 prdstopn 21652 ustneism 22247 itg1addlem4 23686 itg1addlem5 23687 tdeglem4 24040 aannenlem3 24305 efifo 24514 konigsbergssiedgw 27430 pjoml4i 28786 5oai 28860 3oai 28867 bdopssadj 29280 xrge00 30026 xrge0mulc1cn 30327 esumdivc 30485 rpsqrtcn 31011 subfacp1lem5 31504 filnetlem3 32712 filnetlem4 32713 mblfinlem4 33781 itg2gt0cn 33796 psubspset 35551 psubclsetN 35743 relexpaddss 38534 corcltrcl 38555 relopabVD 39657 cncfiooicc 40620 amgmwlem 43074 |
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