Theorem sb4av 2269

Description: Version of sb4a 2501 with a disjoint variable condition, which does not require ax-13 2393. The distinctor antecedent from sb4b 2496 is replaced by a disjoint variable condition in this theorem. (Contributed by NM, 2-Feb-2007.) (Revised by BJ, 15-Dec-2023.)

Assertion

Ref	Expression
sb4av	⊢ ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))

Distinct variable group:   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem sb4av

Step	Hyp	Ref	Expression
1		sp 2208	. . 3 ⊢ (∀𝑡𝜑 → 𝜑)
2	1	sbimi 2097	. 2 ⊢ ([𝑡 / 𝑥]∀𝑡𝜑 → [𝑡 / 𝑥]𝜑)
3		sb6 2108	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
4	2, 3	sylib 220	1 ⊢ ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡 → 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  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-12 2202

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-sb 2081

This theorem is referenced by:  bj-hbsb2av  37237