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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-cbvalnaed | Structured version Visualization version GIF version | ||
| Description: wl-cbvalnae 37534 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.) | 
| Ref | Expression | 
|---|---|
| wl-cbvalnaed.1 | ⊢ Ⅎ𝑥𝜑 | 
| wl-cbvalnaed.2 | ⊢ Ⅎ𝑦𝜑 | 
| wl-cbvalnaed.3 | ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓)) | 
| wl-cbvalnaed.4 | ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) | 
| wl-cbvalnaed.5 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | 
| Ref | Expression | 
|---|---|
| wl-cbvalnaed | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | wl-cbvalnaed.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 2 | wl-cbvalnaed.2 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 3 | wl-cbvalnaed.5 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
| 4 | 1, 2, 3 | wl-dral1d 37532 | . . 3 ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))) | 
| 5 | 4 | imp 406 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | 
| 6 | nfnae 2439 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
| 7 | 1, 6 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) | 
| 8 | wl-nfnae1 37529 | . . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
| 9 | 2, 8 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) | 
| 10 | wl-cbvalnaed.3 | . . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓)) | |
| 11 | 10 | imp 406 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑦𝜓) | 
| 12 | wl-cbvalnaed.4 | . . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) | |
| 13 | 12 | imp 406 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜒) | 
| 14 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | 
| 15 | 7, 9, 11, 13, 14 | cbv2 2408 | . 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | 
| 16 | 5, 15 | pm2.61dan 813 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 Ⅎwnf 1783 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: wl-cbvalnae 37534 | 
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