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Mirrors > Home > MPE Home > Th. List > elabf | Structured version Visualization version GIF version |
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
elabf.1 | ⊢ Ⅎ𝑥𝜓 |
elabf.2 | ⊢ 𝐴 ∈ V |
elabf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elabf | ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabf.2 | . 2 ⊢ 𝐴 ∈ V | |
2 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | elabf.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | elabf.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 2, 3, 4 | elabgf 3675 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 {cab 2712 Vcvv 3478 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-v 3480 |
This theorem is referenced by: elabOLD 3682 dfon2lem1 35765 sdclem2 37729 sdclem1 37730 scottabf 44236 |
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