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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 2825 | . 2 ⊢ 𝐴 ∈ 𝐷 |
5 | 1, 4 | eqeltri 2822 | 1 ⊢ 𝐶 ∈ 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1775 df-cleq 2718 df-clel 2803 |
This theorem is referenced by: oancom 9694 0r 11123 1sr 11124 m1r 11125 smndex1ibas 18890 recvs 25164 recvsOLD 25165 qcvs 25166 wlk2v2elem1 30088 konigsbergiedgw 30181 lmxrge0 33767 brsigarn 34017 ex-sategoelel12 35255 sinccvglem 35500 bj-minftyccb 36932 omcl3g 43000 fouriersw 45852 |
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