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Theorem wl-sb6rft 33460
Description: A specialization of wl-equsal1t 33457. Closed form of sb6rf 2451. (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 2071 . . 3 𝑥𝑥𝜑
2 id 22 . . 3 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3 sbequ12r 2150 . . . 4 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
43a1i 11 . . 3 (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑)))
51, 2, 4wl-equsald 33455 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ 𝜑))
65bicomd 213 1 (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wnf 1748  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by: (None)
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