|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > elabgf | Structured version Visualization version GIF version | ||
| Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.) | 
| Ref | Expression | 
|---|---|
| elabgf.1 | ⊢ Ⅎ𝑥𝐴 | 
| elabgf.2 | ⊢ Ⅎ𝑥𝜓 | 
| elabgf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| elabgf | ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elabgf.1 | . 2 ⊢ Ⅎ𝑥𝐴 | |
| 2 | nfab1 2906 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
| 3 | 1, 2 | nfel 2919 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑} | 
| 4 | elabgf.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 5 | 3, 4 | nfbi 1902 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) | 
| 6 | eleq1 2828 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
| 7 | elabgf.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 8 | 6, 7 | bibi12d 345 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))) | 
| 9 | abid 2717 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
| 10 | 1, 5, 8, 9 | vtoclgf 3568 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 {cab 2713 Ⅎwnfc 2889 | 
| 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: elabf 3674 elab3gf 3683 elrabf 3687 currysetlem 36947 currysetlem1 36949 | 
| Copyright terms: Public domain | W3C validator |