Theorem dfsbimp 2082

Description: A simple consequence of df-sb 2081. (Contributed by Wolf Lammen, 4-Jun-2026.)

Assertion

Ref	Expression
dfsbimp	⊢ ([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 dfsbimp

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sb 2081	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ∧ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))))
2	1	simplbi 499	1 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4  ∀wal 1548  [wsb 2080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399  df-sb 2081

This theorem is referenced by:  dfsb  2083  sbi1lem  2092  spsbe  2105  sbequ2  2274