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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 2904 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 3 | elabf.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 4 | elabf.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 5 | 2, 3, 4 | elabgf 3673 | . 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 1539 Ⅎwnf 1782 ∈ wcel 2107 {cab 2713 Vcvv 3479 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-v 3481 | 
| This theorem is referenced by: dfon2lem1 35785 sdclem2 37750 sdclem1 37751 scottabf 44264 | 
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