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Theorem wl-cbvalnae 34305
 Description: A more general version of cbval 2372 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 2427, nfsb2 2476 or dveeq1 2353. (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 1786 . . 3 𝑥
2 nftru 1786 . . 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 34304 . 2 (⊤ → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
109mptru 1529 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207  ∀wal 1520  ⊤wtru 1523  Ⅎwnf 1765 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766 This theorem is referenced by: (None)
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