Theorem wl-sbid2ft 38122

Description: A more general version of sbid2vw 2301. (Contributed by Wolf Lammen, 14-May-2019.)

Assertion

Ref	Expression
wl-sbid2ft	⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-sbid2ft

Step	Hyp	Ref	Expression
1		sb6 2125	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))
2		nfnf1 2195	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
3		id 23	. . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
4		sbequ12r 2294	. . . 4 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
5	4	a1i 11	. . 3 ⊢ (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑)))
6	2, 3, 5	wl-equsaldv 38117	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ 𝜑))
7	1, 6	bitrid 286	1 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565  Ⅎwnf 1810  [wsb 2097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098

This theorem is referenced by: (None)