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Mirrors > Home > MPE Home > Th. List > elab3 | Structured version Visualization version GIF version |
Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) |
Ref | Expression |
---|---|
elab3.1 | ⊢ (𝜓 → 𝐴 ∈ V) |
elab3.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elab3 | ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elab3.1 | . 2 ⊢ (𝜓 → 𝐴 ∈ V) | |
2 | elab3.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | elab3g 3673 | . 2 ⊢ ((𝜓 → 𝐴 ∈ V) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 {cab 2799 Vcvv 3495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 |
This theorem is referenced by: fvelrnb 6721 elrnmpo 7281 ovelrn 7318 isfi 8527 isnum2 9368 pm54.43lem 9422 isfin3 9712 isfin5 9715 isfin6 9716 genpelv 10416 iswrd 13857 4sqlem2 16279 vdwapval 16303 isghm 18352 issrng 19615 lspsnel 19769 lspprel 19860 iscss 20821 ellspd 20940 istps 21536 islp 21742 is2ndc 22048 elpt 22174 itg2l 24324 elply 24779 isismt 26314 bj-ififc 33910 isline 36869 ispointN 36872 ispsubsp 36875 ispsubclN 37067 islaut 37213 ispautN 37229 istendo 37890 rngunsnply 39766 |
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