Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-cbvalnae | Structured version Visualization version GIF version |
Description: A more general version of cbval 2398 when nonfree properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2448, nfsb2 2487 or dveeq1 2380. (Contributed by Wolf Lammen, 4-Jun-2019.) |
Ref | Expression |
---|---|
wl-cbvalnae.1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑) |
wl-cbvalnae.2 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓) |
wl-cbvalnae.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
wl-cbvalnae | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1807 | . . 3 ⊢ Ⅎ𝑥⊤ | |
2 | nftru 1807 | . . 3 ⊢ Ⅎ𝑦⊤ | |
3 | wl-cbvalnae.1 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑) | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)) |
5 | wl-cbvalnae.2 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)) |
7 | wl-cbvalnae.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))) |
9 | 1, 2, 4, 6, 8 | wl-cbvalnaed 35691 | . 2 ⊢ (⊤ → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
10 | 9 | mptru 1546 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1537 ⊤wtru 1540 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 |
This theorem is referenced by: (None) |
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