Mathbox for Wolf Lammen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-cbvalnaed Structured version   Visualization version   GIF version

Theorem wl-cbvalnaed 32498
 Description: wl-cbvalnae 32499 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.)
Hypotheses
Ref Expression
wl-cbvalnaed.1 𝑥𝜑
wl-cbvalnaed.2 𝑦𝜑
wl-cbvalnaed.3 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓))
wl-cbvalnaed.4 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))
wl-cbvalnaed.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
wl-cbvalnaed (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem wl-cbvalnaed
StepHypRef Expression
1 wl-cbvalnaed.1 . . . 4 𝑥𝜑
2 wl-cbvalnaed.2 . . . 4 𝑦𝜑
3 wl-cbvalnaed.5 . . . 4 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
41, 2, 3wl-dral1d 32497 . . 3 (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))
54imp 444 . 2 ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
6 nfnae 2306 . . . 4 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
71, 6nfan 1816 . . 3 𝑥(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
8 wl-nfnae1 32495 . . . 4 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
92, 8nfan 1816 . . 3 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
10 wl-cbvalnaed.3 . . . 4 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓))
1110imp 444 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑦𝜓)
12 wl-cbvalnaed.4 . . . 4 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))
1312imp 444 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜒)
143adantr 480 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜓𝜒)))
157, 9, 11, 13, 14cbv2 2258 . 2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
165, 15pm2.61dan 828 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by:  wl-cbvalnae  32499
 Copyright terms: Public domain W3C validator