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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-dral1d | Structured version Visualization version GIF version |
Description: A version of dral1 2423 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 𝑥 = 𝑦, ∀𝑥𝑥 = 𝑦 or 𝜑 antecedent. wl-equsal1i 33811 and nf5di 2304 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.) |
Ref | Expression |
---|---|
wl-dral1d.1 | ⊢ Ⅎ𝑥𝜑 |
wl-dral1d.2 | ⊢ Ⅎ𝑦𝜑 |
wl-dral1d.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
wl-dral1d | ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-dral1d.3 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
2 | 1 | com12 32 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒))) |
3 | 2 | pm5.74d 265 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
4 | 3 | sps 2219 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
5 | 4 | dral1 2423 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒))) |
6 | wl-dral1d.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
7 | 6 | 19.21 2241 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
8 | wl-dral1d.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
9 | 8 | 19.21 2241 | . . . 4 ⊢ (∀𝑦(𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦𝜒)) |
10 | 5, 7, 9 | 3bitr3g 305 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑦𝜒))) |
11 | 10 | pm5.74rd 266 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))) |
12 | 11 | com12 32 | 1 ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1651 Ⅎwnf 1879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-10 2185 ax-12 2213 ax-13 2354 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-ex 1876 df-nf 1880 |
This theorem is referenced by: wl-cbvalnaed 33801 |
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