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| Mirrors > Home > MPE Home > Th. List > eleq2w | Structured version Visualization version GIF version | ||
| Description: Weaker version of eleq2 2830 (but more general than elequ2 2123) not depending on ax-ext 2708 nor df-cleq 2729. (Contributed by BJ, 29-Sep-2019.) |
| Ref | Expression |
|---|---|
| eleq2w | ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elequ2 2123 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | |
| 2 | 1 | anbi2d 630 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥) ↔ (𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦))) |
| 3 | 2 | exbidv 1921 | . 2 ⊢ (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦))) |
| 4 | dfclel 2817 | . 2 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥)) | |
| 5 | dfclel 2817 | . 2 ⊢ (𝐴 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦)) | |
| 6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-clel 2816 |
| This theorem is referenced by: clelsb2 2869 eluniab 4921 elintabg 4957 elintabOLD 4959 cantnflem1c 9727 tcrank 9924 isf32lem2 10394 sadcp1 16492 subgacs 19179 nsgacs 19180 sdrgacs 20802 lssacs 20965 elcls3 23091 conncompconn 23440 1stcfb 23453 dfac14lem 23625 r0cld 23746 uffix 23929 flftg 24004 tgpconncompeqg 24120 wilth 27114 tghilberti2 28646 umgr2edgneu 29231 uspgredg2v 29241 usgredgleordALT 29251 nbusgrf1o 29388 vtxdushgrfvedglem 29507 constrmon 33785 ddemeas 34237 cvmcov 35268 cvmseu 35281 sat1el2xp 35384 hilbert1.2 36156 fneint 36349 mnuprdlem1 44291 mnuprdlem2 44292 mnuprdlem4 44294 elunif 45021 fnchoice 45034 lmbr3 45762 |
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