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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 2835 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-cleq 2724 df-clel 2810 |
This theorem is referenced by: eleqtrrid 2840 opth1 5475 opth 5476 eqelsuc 6448 tfrlem11 8390 oalimcl 8562 omlimcl 8580 frgp0 19630 txdis 23143 ordtconnlem1 32973 rankeq1o 35212 |
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