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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 2837 | . 2 ⊢ 𝐴 ∈ 𝐷 |
5 | 1, 4 | eqeltri 2834 | 1 ⊢ 𝐶 ∈ 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-cleq 2729 df-clel 2816 |
This theorem is referenced by: oancom 9263 0r 10691 1sr 10692 m1r 10693 smndex1ibas 18324 recvs 24040 qcvs 24041 wlk2v2elem1 28235 konigsbergiedgw 28328 lmxrge0 31613 brsigarn 31861 ex-sategoelel12 33099 sinccvglem 33340 bj-minftyccb 35128 fouriersw 43445 |
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