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Theorem wl-sb6rft 32509
 Description: A specialization of wl-equsal1t 32506. Closed form of sb6rf 2411. (Contributed by Wolf Lammen, 27-Jul-2019.)
Assertion
Ref Expression
wl-sb6rft (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))

Proof of Theorem wl-sb6rft
StepHypRef Expression
1 nfnf1 2018 . . 3 𝑥𝑥𝜑
2 id 22 . . 3 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3 sbequ12r 2098 . . . 4 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
43a1i 11 . . 3 (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑)))
51, 2, 4wl-equsald 32504 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ 𝜑))
65bicomd 212 1 (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699  [wsb 1867 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-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  df-sb 1868 This theorem is referenced by: (None)
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