Theorem sb6 2124

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 2102. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2197. (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 2103	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		equequ2 2061	. . . . 5 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
3	2	imbi1d 344	. . . 4 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
4	3	albidv 1955	. . 3 ⊢ (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
5	4	equsalvw 2039	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  [wsb 2101

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102

This theorem is used by:  2sb6  2125  sb1v  2126  sbrimvwOLD  2131  sbbiiev  2132  nfs1v  2196  sb4av  2283  sb6a  2296  sb5  2313  sb8v  2385  sb8f  2386  2eu6  2684  nfabdw  2947  elab6g  3626  iota4  6546  axregs  35948  in-ax8  37228  mh-setind  37539  regsfromregtco  37541  regsfromsetind  37542  regsfromunir1  37543  bj-df-sb  37764  bj-dfsbc  37766  bj-ax12ssb  37772  bj-sbievwd  37894  bj-hbs1  37939  bj-hbsb2av  37941  bj-sbievw1  37972  bj-sbievw2  37973  bj-sbievw  37974  wl-sbid2ft  38745  wl-sb9v  38749  wl-lem-moexsb  38768  absnsb  48370  ichnfimlem  48818