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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 2832 | . 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷) | 
| 3 | eleq1i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
| 4 | 3 | eleq1i 2831 | . 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 1539 ∈ wcel 2107 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-cleq 2728 df-clel 2815 | 
| This theorem is referenced by: sbcel12 4410 smndex1n0mnd 18926 zclmncvs 25183 gausslemma2dlem4 27414 bnj98 34882 elmpst 35542 elmpps 35579 sbceqbii 36193 cbvsbcvw2 36232 oaordnrex 43313 omnord1ex 43322 oenord1ex 43333 wfaxpow 45019 unirnmapsn 45224 | 
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