Theorem dfsb 2096

Description: Simplify definition df-sb 2094 by proving the renaming independency. (Contributed by Wolf Lammen, 5-Feb-2026.) df-sb 2094 changed. (Revised by Wolf Lammen, 4-Jun-2026.)

Assertion

Ref	Expression
dfsb	⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable groups:   𝑥,𝑦   𝑦,𝑡   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem dfsb

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbimp 2095	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		df-sb 2094	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ∧ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))))
3		rename-sb 2092	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4	2, 3	just3-df 2091	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → [𝑡 / 𝑥]𝜑)
5	1, 4	impbii 212	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561  [wsb 2093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-sb 2094

This theorem is referenced by:  stdpc4ALT  2102  sbi1ALT  2107  sbequ  2119  sb6  2121  sbal  2206  sbequ1  2286  dfsb7  2316  sbn  2317  sbrim  2341  cbvsbvf  2397  sb4b  2509  sbequbidv  36587  cbvsbdavw  36627  cbvsbdavw2  36628  bj-ssbeq  37137  bj-ssbid2ALT  37147  bj-ssbid1ALT  37149