![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eleq12i | GIF version |
Description: Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
Ref | Expression |
---|---|
eleq1i.1 | ⊢ 𝐴 = 𝐵 |
eleq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
eleq12i | ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
2 | 1 | eleq2i 2104 | . 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷) |
3 | eleq1i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
4 | 3 | eleq1i 2103 | . 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷) |
5 | 2, 4 | bitri 173 | 1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 ∈ wcel 1393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-clel 2036 |
This theorem is referenced by: 3eltr3g 2122 3eltr4g 2123 sbcel12g 2865 |
Copyright terms: Public domain | W3C validator |