Theorem sb6 2119

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 2511). Theorem sb6f 2529 replaces the disjoint variable condition with a nonfreeness hypothesis. Theorem sb4b 2507 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise df-sb 2097. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2192. (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 2098	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		equequ2 2056	. . . . 5 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
3	2	imbi1d 344	. . . 4 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
4	3	albidv 1950	. . 3 ⊢ (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
5	4	equsalvw 2034	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
6	1, 5	bitri 278	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568  [wsb 2096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097

This theorem is referenced by:  2sb6  2120  sb1v  2121  sbrimvwOLD  2126  sbbiiev  2127  sbievwOLD  2129  nfs1v  2191  sb4av  2280  sb6a  2294  sb5  2311  sbievOLD  2348  sb8v  2385  sb8f  2386  2eu6  2684  nfabdw  2946  elab6g  3628  iota4  6517  axregs  35552  in-ax8  36756  mh-setind  37067  regsfromregtco  37069  regsfromsetind  37070  regsfromunir1  37071  bj-df-sb  37292  bj-dfsbc  37294  bj-ax12ssb  37300  bj-sbievwd  37422  bj-hbs1  37467  bj-hbsb2av  37469  bj-sbievw1  37500  bj-sbievw2  37501  bj-sbievw  37502  wl-sbid2ft  38220  wl-sb9v  38224  wl-lem-moexsb  38243  absnsb  47784  ichnfimlem  48232