Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 3eltr3g | Structured version Visualization version GIF version |
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
Ref | Expression |
---|---|
3eltr3g.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
3eltr3g.2 | ⊢ 𝐴 = 𝐶 |
3eltr3g.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3eltr3g | ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eltr3g.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
2 | 3eltr3g.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | 1, 2 | eqeltrrid 2844 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
4 | 3eltr3g.3 | . 2 ⊢ 𝐵 = 𝐷 | |
5 | 3, 4 | eleqtrdi 2849 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-cleq 2730 df-clel 2816 |
This theorem is referenced by: rankelpr 9631 isf34lem7 10135 rmulccn 31878 xrge0mulc1cn 31891 esumpfinvallem 32042 fourierdlem62 43709 |
Copyright terms: Public domain | W3C validator |