Theorem sbt 2102

Description: A substitution into a theorem yields a theorem. See sbtALT 2107 for a shorter proof requiring more axioms. See chvar 2433 and chvarv 2434 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 2098. (Revised by Steven Nguyen, 6-Jul-2023.) Revise df-sb 2098 again. (Revised by Wolf Lammen, 4-Jun-2026.)

Hypothesis

Ref	Expression
sbtlem.1	⊢ 𝜑

Assertion

Ref	Expression
sbt	⊢ [𝑡 / 𝑥]𝜑

Proof of Theorem sbt

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbtlem.1	. . . 4 ⊢ 𝜑
2	1	sbtlem 2101	. . 3 ⊢ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1	sbtlem 2101	. . 3 ⊢ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))
4	2, 3	pm3.2i 475	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ∧ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
5		df-sb 2098	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ∧ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))))
6	4, 5	mpbir 234	1 ⊢ [𝑡 / 𝑥]𝜑

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565  [wsb 2097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822

This theorem depends on definitions:  df-bi 210  df-an 401  df-sb 2098

This theorem is referenced by:  sbtru  2103  sbimi  2114  vexw  2753  iscatd2  17737  iuninc  32846  suppss2f  32924  esumpfinvalf  34411  sbtT  45168  2sb5ndVD  45510  2sb5ndALT  45532  icht  48090