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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-cbvalnaed | Structured version Visualization version GIF version |
Description: wl-cbvalnae 35347 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 35345 | . . 3 ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))) |
5 | 4 | imp 410 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
6 | nfnae 2435 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
7 | 1, 6 | nfan 1906 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
8 | wl-nfnae1 35342 | . . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
9 | 2, 8 | nfan 1906 | . . 3 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
10 | wl-cbvalnaed.3 | . . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓)) | |
11 | 10 | imp 410 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑦𝜓) |
12 | wl-cbvalnaed.4 | . . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) | |
13 | 12 | imp 410 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜒) |
14 | 3 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
15 | 7, 9, 11, 13, 14 | cbv2 2404 | . 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 209 ∧ wa 399 ∀wal 1540 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-11 2162 ax-12 2179 ax-13 2373 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 |
This theorem is referenced by: wl-cbvalnae 35347 |
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