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| Mirrors > Home > MPE Home > Th. List > cbvexeqsetf | Structured version Visualization version GIF version | ||
| Description: The expression ∃𝑥𝑥 = 𝐴 means "𝐴 is a set" even if 𝐴 contains 𝑥 as a bound variable. This lemma helps minimizing axiom or df-clab 2714 usage in some cases. Extracted from the proof of issetft 3495. (Contributed by Wolf Lammen, 30-Jul-2025.) | 
| Ref | Expression | 
|---|---|
| cbvexeqsetf | ⊢ (Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfnfc1 2907 | . . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
| 2 | nfv 1913 | . . . 4 ⊢ Ⅎ𝑦Ⅎ𝑥𝐴 | |
| 3 | nfvd 1914 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴) | |
| 4 | nfcvd 2905 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
| 5 | id 22 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
| 6 | 4, 5 | nfeqd 2915 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) | 
| 7 | 6 | nfnd 1857 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴) | 
| 8 | eqeq1 2740 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
| 9 | 8 | notbid 318 | . . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 = 𝐴 ↔ ¬ 𝑦 = 𝐴)) | 
| 10 | 9 | a1i 11 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → (𝑥 = 𝑦 → (¬ 𝑥 = 𝐴 ↔ ¬ 𝑦 = 𝐴))) | 
| 11 | 1, 2, 3, 7, 10 | cbv2w 2338 | . . 3 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑦 ¬ 𝑦 = 𝐴)) | 
| 12 | alnex 1780 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴) | |
| 13 | alnex 1780 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴) | |
| 14 | 11, 12, 13 | 3bitr3g 313 | . 2 ⊢ (Ⅎ𝑥𝐴 → (¬ ∃𝑥 𝑥 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴)) | 
| 15 | 14 | con4bid 317 | 1 ⊢ (Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∀wal 1537 = wceq 1539 ∃wex 1778 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-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-ex 1779 df-nf 1783 df-cleq 2728 df-nfc 2891 | 
| This theorem is referenced by: issetft 3495 spcimgft 3545 | 
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