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Theorem wl-2sb6d 33674
Description: Version of 2sb6 2277 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
Hypotheses
Ref Expression
wl-2sb6d.1 (𝜑 → ¬ ∀𝑦 𝑦 = 𝑥)
wl-2sb6d.2 (𝜑 → ¬ ∀𝑦 𝑦 = 𝑤)
wl-2sb6d.3 (𝜑 → ¬ ∀𝑦 𝑦 = 𝑧)
wl-2sb6d.4 (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)
Assertion
Ref Expression
wl-2sb6d (𝜑 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓)))

Proof of Theorem wl-2sb6d
StepHypRef Expression
1 wl-2sb6d.4 . 2 (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)
2 wl-2sb6d.2 . 2 (𝜑 → ¬ ∀𝑦 𝑦 = 𝑤)
3 wl-2sb6d.1 . . 3 (𝜑 → ¬ ∀𝑦 𝑦 = 𝑥)
4 wl-2sb6d.3 . . 3 (𝜑 → ¬ ∀𝑦 𝑦 = 𝑧)
53, 4jca 501 . 2 (𝜑 → (¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧))
6 wl-sb6nae 33673 . . 3 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜓)))
7 nfnae 2470 . . . . 5 𝑥 ¬ ∀𝑦 𝑦 = 𝑤
8 wl-nfnae1 33650 . . . . . 6 𝑥 ¬ ∀𝑦 𝑦 = 𝑥
9 nfnae 2470 . . . . . 6 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
108, 9nfan 1980 . . . . 5 𝑥(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
117, 10nfan 1980 . . . 4 𝑥(¬ ∀𝑦 𝑦 = 𝑤 ∧ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧))
12 wl-sb6nae 33673 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑤 → ([𝑤 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑤𝜓)))
1312imbi2d 329 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑤 → ((𝑥 = 𝑧 → [𝑤 / 𝑦]𝜓) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜓))))
14 impexp 437 . . . . . . 7 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜓)))
1514albii 1895 . . . . . 6 (∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜓)))
16 nfeqf 2457 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑥 = 𝑧)
17 19.21t 2229 . . . . . . 7 (Ⅎ𝑦 𝑥 = 𝑧 → (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜓)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜓))))
1816, 17syl 17 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜓)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜓))))
1915, 18syl5rbb 273 . . . . 5 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ((𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜓)) ↔ ∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓)))
2013, 19sylan9bb 499 . . . 4 ((¬ ∀𝑦 𝑦 = 𝑤 ∧ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)) → ((𝑥 = 𝑧 → [𝑤 / 𝑦]𝜓) ↔ ∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓)))
2111, 20albid 2246 . . 3 ((¬ ∀𝑦 𝑦 = 𝑤 ∧ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)) → (∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜓) ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓)))
226, 21sylan9bb 499 . 2 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ (¬ ∀𝑦 𝑦 = 𝑤 ∧ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧))) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓)))
231, 2, 5, 22syl12anc 1474 1 (𝜑 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629  wnf 1856  [wsb 2049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050
This theorem is referenced by:  wl-sbcom2d-lem2  33676
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