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Theorem wl-cbvalnae 33450
Description: A more general version of cbval 2307 when non-free properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2365, nfsb2 2388 or dveeq1 2336. (Contributed by Wolf Lammen, 4-Jun-2019.)
Hypotheses
Ref Expression
wl-cbvalnae.1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)
wl-cbvalnae.2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
wl-cbvalnae.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
wl-cbvalnae (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Proof of Theorem wl-cbvalnae
StepHypRef Expression
1 nftru 1770 . . 3 𝑥
2 nftru 1770 . . 3 𝑦
3 wl-cbvalnae.1 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)
43a1i 11 . . 3 (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑))
5 wl-cbvalnae.2 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
65a1i 11 . . 3 (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓))
7 wl-cbvalnae.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
87a1i 11 . . 3 (⊤ → (𝑥 = 𝑦 → (𝜑𝜓)))
91, 2, 4, 6, 8wl-cbvalnaed 33449 . 2 (⊤ → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
109trud 1533 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1521  wtru 1524  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
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