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Mirrors > Home > ILE Home > Th. List > eleqtrd | GIF version |
Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
Ref | Expression |
---|---|
eleqtrd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
eleqtrd.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
eleqtrd | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleqtrd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | eleqtrd.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
3 | 2 | eleq2d 2158 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)) |
4 | 1, 3 | mpbid 146 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 ∈ wcel 1439 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-4 1446 ax-17 1465 ax-ial 1473 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-cleq 2082 df-clel 2085 |
This theorem is referenced by: eleqtrrd 2168 3eltr3d 2171 syl5eleq 2177 syl6eleq 2181 opth1 4072 0nelop 4084 tfisi 4415 ercl 6317 erth 6350 ecelqsdm 6376 phpm 6635 lincmb01cmp 9481 fzopth 9536 fzoaddel2 9665 fzosubel2 9667 fzocatel 9671 zpnn0elfzo1 9680 fzoend 9694 peano2fzor 9704 monoord2 9966 isermono 9967 bcpasc 10235 zfz1isolemiso 10305 fisum0diag2 10902 isumsplit 10946 |
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