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| Mirrors > Home > MPE Home > Th. List > eleq12 | Structured version Visualization version GIF version | ||
| Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.) |
| Ref | Expression |
|---|---|
| eleq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq1 2829 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
| 2 | eleq2 2830 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | |
| 3 | 1, 2 | sylan9bb 509 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 |
| This theorem is referenced by: rru 3785 trel 5268 epelg 5585 preleqg 9655 preleqALT 9657 oemapval 9723 cantnf 9733 wemapwe 9737 nnsdomel 10030 cldval 23031 isufil 23911 taylthlem2 26416 umgr2v2enb1 29544 issiga 34113 bj-epelg 37069 rdgssun 37379 matunitlindf 37625 wepwsolem 43054 aomclem8 43073 grumnud 44305 nelbr 47286 |
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