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Theorem sbt 2071
 Description: A substitution into a theorem yields a theorem. See sbtALT 2074 for a shorter proof requiring more axioms. See chvar 2402 and chvarv 2403 for versions using implicit substitution. (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Jul-2018.) Revise df-sb 2070. (Revised by Steven Nguyen, 6-Jul-2023.)
Hypothesis
Ref Expression
sbt.1 𝜑
Assertion
Ref Expression
sbt [𝑡 / 𝑥]𝜑

Proof of Theorem sbt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbt.1 . . . . . 6 𝜑
21a1i 11 . . . . 5 (𝑥 = 𝑦𝜑)
32ax-gen 1797 . . . 4 𝑥(𝑥 = 𝑦𝜑)
43a1i 11 . . 3 (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑))
54ax-gen 1797 . 2 𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑))
6 df-sb 2070 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
75, 6mpbir 234 1 [𝑡 / 𝑥]𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797 This theorem depends on definitions:  df-bi 210  df-sb 2070 This theorem is referenced by:  sbtru  2072  sbimi  2079  vexw  2782  vjust  3442  iscatd2  16947  iuninc  30334  suppss2f  30408  esumpfinvalf  31460  wl-rgenw  35027  sbtT  41316  2sb5ndVD  41659  2sb5ndALT  41681  icht  44012
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