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Theorem wl-cbvalnaed 33801
Description: wl-cbvalnae 33802 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 33800 . . 3 (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))
54imp 396 . 2 ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
6 nfnae 2418 . . . 4 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
71, 6nfan 1999 . . 3 𝑥(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
8 wl-nfnae1 33798 . . . 4 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
92, 8nfan 1999 . . 3 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
10 wl-cbvalnaed.3 . . . 4 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓))
1110imp 396 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑦𝜓)
12 wl-cbvalnaed.4 . . . 4 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))
1312imp 396 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜒)
143adantr 473 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜓𝜒)))
157, 9, 11, 13, 14cbv2 2386 . 2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
165, 15pm2.61dan 848 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wal 1651  wnf 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880
This theorem is referenced by:  wl-cbvalnae  33802
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