Theorem sb5 2266

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 2089 and even needs no disjoint variable condition, see sb1 2476. Theorem sb5f 2496 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 2087	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sbalex 2234	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	bitr4i 278	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1538  ∃wex 1780  [wsb 2066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1781  df-nf 1785  df-sb 2067

This theorem is referenced by:  sb56OLD  2268  2sb5  2270  sb7f  2523  clelab  2878  sbc2or  3786  sbc5ALT  3806