![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eleqtrid | Structured version Visualization version GIF version |
Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
Ref | Expression |
---|---|
eleqtrid.1 | ⊢ 𝐴 ∈ 𝐵 |
eleqtrid.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
eleqtrid | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleqtrid.1 | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
3 | eleqtrid.2 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) | |
4 | 2, 3 | eleqtrd 2892 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: eleqtrrid 2897 opth1 5332 opth 5333 eqelsuc 6240 tfrlem11 8007 oalimcl 8169 omlimcl 8187 frgp0 18878 txdis 22237 ordtconnlem1 31277 rankeq1o 33745 |
Copyright terms: Public domain | W3C validator |