Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbtd Structured version   Visualization version   GIF version

Theorem sbtd 39390
Description: A true statement is true upon substitution (deduction). A similar proof is possible for icht 43966. (Contributed by SN, 4-May-2024.)
Hypothesis
Ref Expression
sbtd.1 (𝜑𝜓)
Assertion
Ref Expression
sbtd (𝜑 → [𝑡 / 𝑥]𝜓)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑡)   𝜓(𝑥,𝑡)

Proof of Theorem sbtd
StepHypRef Expression
1 sbtd.1 . . 3 (𝜑𝜓)
21alrimiv 1928 . 2 (𝜑 → ∀𝑥𝜓)
3 stdpc4 2073 . 2 (∀𝑥𝜓 → [𝑡 / 𝑥]𝜓)
42, 3syl 17 1 (𝜑 → [𝑡 / 𝑥]𝜓)
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  ax-4 1811  ax-5 1911
This theorem depends on definitions:  df-bi 210  df-sb 2070
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator