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Mirrors > Home > MPE Home > Th. List > 3eltr3i | Structured version Visualization version GIF version |
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
3eltr3i.1 | ⊢ 𝐴 ∈ 𝐵 |
3eltr3i.2 | ⊢ 𝐴 = 𝐶 |
3eltr3i.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3eltr3i | ⊢ 𝐶 ∈ 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eltr3i.2 | . 2 ⊢ 𝐴 = 𝐶 | |
2 | 3eltr3i.1 | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
3 | 3eltr3i.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
4 | 2, 3 | eleqtri 2836 | . 2 ⊢ 𝐴 ∈ 𝐷 |
5 | 1, 4 | eqeltrri 2835 | 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: raddcn 31590 clsk1independent 41331 fourierdlem62 43382 |
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