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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 2872 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
2 | eleq2 2873 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | |
3 | 1, 2 | sylan9bb 510 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1766 df-cleq 2790 df-clel 2865 |
This theorem is referenced by: rru 3709 trel 5077 epelg 5361 epelgOLD 5362 preleqg 8931 preleqALT 8933 oemapval 8999 cantnf 9009 wemapwe 9013 nnsdomel 9272 cldval 21319 isufil 22199 umgr2v2enb1 26995 issiga 30984 bj-elep 33979 rdgssun 34211 fvineqsneu 34244 matunitlindf 34442 wepwsolem 39148 aomclem8 39167 grumnud 40140 nelbr 43011 |
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