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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 2831 | . 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷) |
3 | eleq1i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
4 | 3 | eleq1i 2830 | . 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷) |
5 | 2, 4 | bitri 275 | 1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-clel 2814 |
This theorem is referenced by: sbcel12 4417 smndex1n0mnd 18938 zclmncvs 25196 gausslemma2dlem4 27428 bnj98 34860 elmpst 35521 elmpps 35558 sbceqbii 36173 cbvsbcvw2 36213 oaordnrex 43285 omnord1ex 43294 oenord1ex 43305 unirnmapsn 45157 |
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