Theorem sb6 2091

Description: Alternate definition of substitution when variables are disjoint. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" also holds without a disjoint variable condition (sb2 2484). Theorem sb6f 2502 replaces the disjoint variable condition with a nonfreeness hypothesis. Theorem sb4b 2480 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise df-sb 2069. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2163. (Revised by Steven Nguyen, 7-Jul-2023.) (Proof shortened by Wolf Lammen, 16-Jul-2023.)

Assertion

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

Distinct variable group:   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem sb6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsb 2070	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		equequ2 2028	. . . . 5 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
3	2	imbi1d 341	. . . 4 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
4	3	albidv 1922	. . 3 ⊢ (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
5	4	equsalvw 2006	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
6	1, 5	bitri 275	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069

This theorem is referenced by:  2sb6  2092  sb1v  2093  sbrimvw  2097  sbbiiev  2098  sbievwOLD  2100  nfs1v  2162  sb4av  2252  sb6a  2266  sb5  2283  sbievOLD  2321  sb8v  2358  sb8f  2359  2eu6  2658  nfabdw  2921  elab6g  3625  iota4  6481  axregs  35317  in-ax8  36440  mh-setind  36688  regsfromregtr  36690  regsfromsetind  36691  regsfromunir1  36692  bj-df-sb  36897  bj-ax12ssb  36903  bj-sbievwd  37017  bj-hbs1  37060  bj-hbsb2av  37062  bj-sbievw1  37093  bj-sbievw2  37094  bj-sbievw  37095  wl-sbid2ft  37800  wl-sb9v  37804  wl-lem-moexsb  37823  absnsb  47387  ichnfimlem  47823