MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb6 Structured version   Visualization version   GIF version

Theorem sb6 2083
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 2482). Theorem sb6f 2500 replaces the disjoint variable condition with a nonfreeness hypothesis. Theorem sb4b 2478 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise df-sb 2063. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2155. (Revised by Steven Nguyen, 7-Jul-2023.) (Proof shortened by Wolf Lammen, 16-Jul-2023.)
Assertion
Ref Expression
sb6 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem sb6
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-sb 2063 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 equequ2 2023 . . . . 5 (𝑦 = 𝑡 → (𝑥 = 𝑦𝑥 = 𝑡))
32imbi1d 341 . . . 4 (𝑦 = 𝑡 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
43albidv 1918 . . 3 (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
54equsalvw 2001 . 2 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑥(𝑥 = 𝑡𝜑))
61, 5bitri 275 1 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063
This theorem is referenced by:  2sb6  2084  sb1v  2085  sbrimvw  2089  sbbiiev  2090  sbievwOLD  2092  nfs1v  2154  sb4av  2242  sb6a  2256  sb5  2274  sbievOLD  2314  sb8v  2353  sb8f  2354  2eu6  2655  nfabdw  2925  elab6g  3669  disj  4456  iota4  6544  in-ax8  36207  bj-ax12ssb  36641  bj-sbievwd  36765  bj-hbs1  36795  bj-hbsb2av  36797  bj-sbievw1  36828  bj-sbievw2  36829  bj-sbievw  36830  wl-sbid2ft  37526  wl-sb9v  37530  wl-lem-moexsb  37549  absnsb  46977  ichnfimlem  47388
  Copyright terms: Public domain W3C validator