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Mirrors > Home > NFE Home > Th. List > eleq12d | GIF version |
Description: Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
Ref | Expression |
---|---|
eleq1d.1 | ⊢ (φ → A = B) |
eleq12d.2 | ⊢ (φ → C = D) |
Ref | Expression |
---|---|
eleq12d | ⊢ (φ → (A ∈ C ↔ B ∈ D)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq12d.2 | . . 3 ⊢ (φ → C = D) | |
2 | 1 | eleq2d 2420 | . 2 ⊢ (φ → (A ∈ C ↔ A ∈ D)) |
3 | eleq1d.1 | . . 3 ⊢ (φ → A = B) | |
4 | 3 | eleq1d 2419 | . 2 ⊢ (φ → (A ∈ D ↔ B ∈ D)) |
5 | 2, 4 | bitrd 244 | 1 ⊢ (φ → (A ∈ C ↔ B ∈ D)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-cleq 2346 df-clel 2349 |
This theorem is referenced by: cbvraldva2 2839 cbvrexdva2 2840 ru 3045 sbcel12g 3151 cbvralcsf 3198 cbvreucsf 3200 cbvrabcsf 3201 nenpw1pwlem2 6085 nmembers1 6271 |
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