Theorem sb4v 2248

Description: Version of sb4 2431 with a disjoint variable condition instead of a distinctor antecedent, which does not require ax-13 2333. (Contributed by BJ, 23-Jun-2019.)

Assertion

Ref	Expression
sb4v	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb4v

Step	Hyp	Ref	Expression
1		sb1 2014	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		sb56 2247	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	sylib 210	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1599  ∃wex 1823  [wsb 2011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-12 2162

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012

This theorem is referenced by:  sb6  2249  sbi1v  2277  bj-hbs1  33354