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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 3490 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | elab4g.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | elab4g.2 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
4 | 2, 3 | elab2g 3669 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
5 | 1, 4 | biadanii 821 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {cab 2705 Vcvv 3471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 |
This theorem is referenced by: isprs 18289 ispos 18306 istrkgc 28271 istrkgb 28272 istrkgcb 28273 istrkge 28274 istrkgl 28275 eulerpartlemt0 33989 istrkg2d 34298 |
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