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| Mirrors > Home > MPE Home > Th. List > elab4g | Structured version Visualization version GIF version | ||
| Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.) |
| Ref | Expression |
|---|---|
| elab4g.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| elab4g.2 | ⊢ 𝐵 = {𝑥 ∣ 𝜑} |
| Ref | Expression |
|---|---|
| elab4g | ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3501 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
| 2 | elab4g.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 3 | elab4g.2 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
| 4 | 2, 3 | elab2g 3680 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
| 5 | 1, 4 | biadanii 822 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 |
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 |
| This theorem is referenced by: isprs 18342 ispos 18360 istrkgc 28462 istrkgb 28463 istrkgcb 28464 istrkge 28465 istrkgl 28466 eulerpartlemt0 34371 istrkg2d 34681 |
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