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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 2914 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-cleq 2813 df-clel 2892 |
This theorem is referenced by: eleqtrrid 2919 opth1 5364 opth 5365 eqelsuc 6269 tfrlem11 8021 oalimcl 8183 omlimcl 8201 frgp0 18882 txdis 22236 ordtconnlem1 31191 rankeq1o 33656 |
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