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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 3498 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | elab4g.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | elab4g.2 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
4 | 2, 3 | elab2g 3682 | . 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 1536 ∈ wcel 2105 {cab 2711 Vcvv 3477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 |
This theorem is referenced by: isprs 18353 ispos 18371 istrkgc 28476 istrkgb 28477 istrkgcb 28478 istrkge 28479 istrkgl 28480 eulerpartlemt0 34350 istrkg2d 34659 |
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