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Theorem sb4av 2242
Description: Version of sb4a 2483 with a disjoint variable condition, which does not require ax-13 2375. The distinctor antecedent from sb4b 2478 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 = 𝑡𝜑))
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem sb4av
StepHypRef Expression
1 sp 2181 . . 3 (∀𝑡𝜑𝜑)
21sbimi 2072 . 2 ([𝑡 / 𝑥]∀𝑡𝜑 → [𝑡 / 𝑥]𝜑)
3 sb6 2083 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))
42, 3sylib 218 1 ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063
This theorem is referenced by:  bj-hbsb2av  36797
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