![]() |
Mathbox for Wolf Lammen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-dral1d | Structured version Visualization version GIF version |
Description: A version of dral1 2432 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 37199 and nf5di 2274 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 272 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
4 | 3 | sps 2173 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
5 | 4 | dral1 2432 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒))) |
6 | wl-dral1d.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
7 | 6 | 19.21 2195 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
8 | wl-dral1d.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
9 | 8 | 19.21 2195 | . . . 4 ⊢ (∀𝑦(𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦𝜒)) |
10 | 5, 7, 9 | 3bitr3g 312 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑦𝜒))) |
11 | 10 | pm5.74rd 273 | . 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 205 ∀wal 1531 Ⅎwnf 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-12 2166 ax-13 2365 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1774 df-nf 1778 |
This theorem is referenced by: wl-cbvalnaed 37187 |
Copyright terms: Public domain | W3C validator |