Theorem sb6 2088

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 2479). Theorem sb6f 2497 replaces the disjoint variable condition with a nonfreeness hypothesis. Theorem sb4b 2475 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise df-sb 2068. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2160. (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		df-sb 2068	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		equequ2 2027	. . . . 5 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
3	2	imbi1d 341	. . . 4 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
4	3	albidv 1921	. . 3 ⊢ (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
5	4	equsalvw 2005	. 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 1539  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068

This theorem is referenced by:  2sb6  2089  sb1v  2090  sbrimvw  2094  sbbiiev  2095  sbievwOLD  2097  nfs1v  2159  sb4av  2247  sb6a  2261  sb5  2278  sbievOLD  2316  sb8v  2353  sb8f  2354  2eu6  2652  nfabdw  2916  elab6g  3619  iota4  6462  axregs  35145  in-ax8  36268  bj-ax12ssb  36702  bj-sbievwd  36826  bj-hbs1  36856  bj-hbsb2av  36858  bj-sbievw1  36889  bj-sbievw2  36890  bj-sbievw  36891  wl-sbid2ft  37589  wl-sb9v  37593  wl-lem-moexsb  37612  absnsb  47137  ichnfimlem  47573