Theorem sb6 1910

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 1909	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	anbi2i 457	. 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		df-sb 1786	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
4		ax-4 1533	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
5	4	pm4.71ri 392	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	2, 3, 5	3bitr4i 212	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371  ∃wex 1515  [wsb 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-sb 1786

This theorem is referenced by:  sb5  1911  sbnv  1912  sbanv  1913  sbi1v  1915  sbi2v  1916  hbs1  1966  2sb6  2012  sbcom2v  2013  sb6a  2016  sb7af  2021  sbalyz  2027  sbal1yz  2029  exsb  2036  sbal2  2048  cbvabw  2328  nfabdw  2367  csbcow  3104