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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 2858 | . 2 ⊢ 𝐴 ∈ 𝐷 |
5 | 1, 4 | eqeltri 2855 | 1 ⊢ 𝐶 ∈ 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1508 ∈ wcel 2051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1744 df-cleq 2764 df-clel 2839 |
This theorem is referenced by: oancom 8906 0r 10298 1sr 10299 m1r 10300 recvs 23468 qcvs 23469 wlk2v2elem1 27699 konigsbergiedgw 27795 lmxrge0 30871 brsigarn 31120 sinccvglem 32472 bj-minftyccb 34013 fouriersw 41981 |
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