Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eleq12i | Structured version Visualization version 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 2901 | . 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷) |
3 | eleq1i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
4 | 3 | eleq1i 2900 | . 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷) |
5 | 2, 4 | bitri 276 | 1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-clel 2890 |
This theorem is referenced by: sbcel12 4357 zclmncvs 23679 gausslemma2dlem4 25872 bnj98 32038 elmpst 32680 elmpps 32717 unirnmapsn 41353 smndex1n0mnd 44012 |
Copyright terms: Public domain | W3C validator |