Theorem sb5 2313

Description: Alternate definition of substitution when variables are disjoint. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1v 2124 and even needs no disjoint variable condition, see sb1 2512. Theorem sb5f 2532 replaces the disjoint variable condition with a nonfreeness hypothesis. (Contributed by NM, 18-Aug-1993.) (Revised by Wolf Lammen, 4-Sep-2023.)

Assertion

Ref	Expression
sb5	⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem sb5

Step	Hyp	Ref	Expression
1		sb6 2122	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sbalex 2281	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	bitr4i 281	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  2sb5  2315  sb7f  2559  clelab  2909  sbc2or  3755  sbc5ALT  3775  bj-axseprep  37768