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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-cbvalnaed | Structured version Visualization version GIF version |
Description: wl-cbvalnae 33320 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 33318 | . . 3 ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))) |
5 | 4 | imp 445 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
6 | nfnae 2318 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
7 | 1, 6 | nfan 1828 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
8 | wl-nfnae1 33316 | . . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
9 | 2, 8 | nfan 1828 | . . 3 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
10 | wl-cbvalnaed.3 | . . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓)) | |
11 | 10 | imp 445 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑦𝜓) |
12 | wl-cbvalnaed.4 | . . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) | |
13 | 12 | imp 445 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜒) |
14 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
15 | 7, 9, 11, 13, 14 | cbv2 2270 | . 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
16 | 5, 15 | pm2.61dan 832 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 |
This theorem is referenced by: wl-cbvalnae 33320 |
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