Theorem dfsb 2070

Description: Simplify definition df-sb 2069 by removing its provable hypothesis. (Contributed by Wolf Lammen, 5-Feb-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		sbjust 2067	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
2	1	df-sb 2069	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069

This theorem is referenced by:  stdpc4  2074  sbi1  2077  spsbe  2088  sbequ  2089  sb6  2091  sbal  2175  hbsbwOLD  2178  sbequ1  2256  sbequ2  2257  dfsb7  2286  sbn  2287  sbrim  2311  cbvsbvf  2368  sb4b  2480  sbequbidv  36389  cbvsbdavw  36429  cbvsbdavw2  36430  bj-ssbeq  36829  bj-ssbid2ALT  36839  bj-ssbid1ALT  36841