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

Step	Hyp	Ref	Expression
1		nfnf1 2018	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
2		id 22	. . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3		sbequ12r 2098	. . . 4 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
4	3	a1i 11	. . 3 ⊢ (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑)))
5	1, 2, 4	wl-equsald 32504	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ 𝜑))
6	5	bicomd 212	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))

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)