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Theorem sbt 2062
Description: A substitution into a theorem yields a theorem. See sbtALT 2065 for a shorter proof requiring more axioms. See chvar 2404 and chvarv 2405 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 2061. (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 1787 . . . 4 𝑥(𝑥 = 𝑦𝜑)
43a1i 11 . . 3 (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑))
54ax-gen 1787 . 2 𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑))
6 df-sb 2061 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
75, 6mpbir 232 1 [𝑡 / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787
This theorem depends on definitions:  df-bi 208  df-sb 2061
This theorem is referenced by:  sbtru  2063  sbimi  2070  vexw  2802  vjust  3493  iscatd2  16940  iuninc  30240  suppss2f  30312  esumpfinvalf  31234  wl-rgenw  34724  sbtT  40778  2sb5ndVD  41121  2sb5ndALT  41143
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