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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 1601 ∈ wcel 2107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1824 df-cleq 2770 df-clel 2774 |
This theorem is referenced by: oancom 8845 0r 10237 1sr 10238 m1r 10239 recvs 23353 qcvs 23354 wlk2v2elem1 27558 konigsbergiedgw 27654 lmxrge0 30596 brsigarn 30845 sinccvglem 32163 bj-minftyccb 33702 fouriersw 41375 |
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