Theorem wl-dv-sb 33820

Description: Substitution for 𝑥 has no effect on 𝜑 not containing 𝑥. See also sbf 2497. (Contributed by Wolf Lammen, 4-Sep-2022.)

Assertion

Ref	Expression
wl-dv-sb	⊢ (𝜑 ↔ [𝑦 / 𝑥]𝜑)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem wl-dv-sb

Step	Hyp	Ref	Expression
1		ax-1 6	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑))
2		ax6e 2390	. . . . . 6 ⊢ ∃𝑥 𝑥 = 𝑦
3	2	jctl 520	. . . . 5 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝑦 ∧ 𝜑))
4		19.41v 2045	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜑))
5	3, 4	sylibr 226	. . . 4 ⊢ (𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
6	1, 5	jca 508	. . 3 ⊢ (𝜑 → ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
7		df-sb 2065	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
8	6, 7	sylibr 226	. 2 ⊢ (𝜑 → [𝑦 / 𝑥]𝜑)
9		spsbe 2068	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
10		ax5e 2008	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
11	9, 10	syl 17	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜑)
12	8, 11	impbii 201	1 ⊢ (𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385  ∃wex 1875  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213  ax-13 2377

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-sb 2065

This theorem is referenced by: (None)