Theorem sb6 1858

Description: Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)

Assertion

Ref	Expression
sb6	⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb6

Step	Hyp	Ref	Expression
1		sb56 1857	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	anbi2i 452	. 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		df-sb 1736	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
4		ax-4 1487	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
5	4	pm4.71ri 389	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	2, 3, 5	3bitr4i 211	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329  ∃wex 1468  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-sb 1736

This theorem is referenced by:  sb5  1859  sbnv  1860  sbanv  1861  sbi1v  1863  sbi2v  1864  hbs1  1909  2sb6  1957  sbcom2v  1958  sb6a  1961  sb7af  1966  sbalyz  1972  sbal1yz  1974  exsb  1981  sbal2  1995