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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 2838 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
4 | 3eltr3g.3 | . 2 ⊢ 𝐵 = 𝐷 | |
5 | 3, 4 | eleqtrdi 2843 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-cleq 2730 df-clel 2811 |
This theorem is referenced by: rankelpr 9375 isf34lem7 9879 rmulccn 31450 xrge0mulc1cn 31463 esumpfinvallem 31612 fourierdlem62 43271 |
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