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| Mirrors > Home > MPE Home > Th. List > 3eltr4i | Structured version Visualization version GIF version | ||
| Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| 3eltr4i.1 | ⊢ 𝐴 ∈ 𝐵 | 
| 3eltr4i.2 | ⊢ 𝐶 = 𝐴 | 
| 3eltr4i.3 | ⊢ 𝐷 = 𝐵 | 
| Ref | Expression | 
|---|---|
| 3eltr4i | ⊢ 𝐶 ∈ 𝐷 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 3eltr4i.2 | . 2 ⊢ 𝐶 = 𝐴 | |
| 2 | 3eltr4i.1 | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
| 3 | 3eltr4i.3 | . . 3 ⊢ 𝐷 = 𝐵 | |
| 4 | 2, 3 | eleqtrri 2840 | . 2 ⊢ 𝐴 ∈ 𝐷 | 
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 𝐶 ∈ 𝐷 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 | 
| This theorem is referenced by: oancom 9691 0r 11120 1sr 11121 m1r 11122 smndex1ibas 18913 recvs 25179 recvsOLD 25180 qcvs 25181 wlk2v2elem1 30174 konigsbergiedgw 30267 lmxrge0 33951 brsigarn 34185 ex-sategoelel12 35432 sinccvglem 35677 bj-minftyccb 37226 resuppsinopn 42393 omcl3g 43347 fouriersw 46246 | 
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