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Theorem wl-sb6rft 34907
Description: A specialization of wl-equsal1t 34904. Closed form of sb6rf 2492. (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 2158 . . 3 𝑥𝑥𝜑
2 id 22 . . 3 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3 sbequ12r 2255 . . . 4 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
43a1i 11 . . 3 (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑)))
51, 2, 4wl-equsald 34902 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ 𝜑))
65bicomd 226 1 (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wnf 1785  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by: (None)
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