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 2480). Theorem sb6f 2501 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 2154. (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 2029	. . . . 5 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
3	2	imbi1d 342	. . . 4 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
4	3	albidv 1923	. . 3 ⊢ (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
5	4	equsalvw 2007	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
6	1, 5	bitri 274	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068

This theorem is referenced by:  2sb6  2089  sb1v  2090  sbrimvw  2094  sbievw  2095  sbcom3vv  2098  nfs1v  2153  sb4av  2236  sb6a  2250  sb5  2268  sb56OLD  2270  sbiev  2309  sb8v  2350  sb8f  2351  2eu6  2658  nfabdw  2930  nfabdwOLD  2931  elab6g  3600  ab0OLD  4309  disj  4381  iota4  6414  bj-ax12ssb  34839  bj-sbievwd  34964  bj-hbs1  34994  bj-hbsb2av  34996  bj-sbievw1  35029  bj-sbievw2  35030  bj-sbievw  35031  wl-lem-moexsb  35723  absnsb  44521  ichnfimlem  44915