Theorem sbt 2105

Description: A substitution into a theorem yields a theorem. See sbtALT 2108 for a shorter proof requiring more axioms. See chvar 2427 and chvarv 2428 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 2102. (Revised by Steven Nguyen, 6-Jul-2023.) Revise df-sb 2102 again. (Revised by Wolf Lammen, 4-Feb-2026.)

Hypothesis

Ref	Expression
sbt.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		sbt.1	. . 3 ⊢ 𝜑
2	1	sbtlem 2104	. 2 ⊢ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1	sbtlem 2104	. . . 4 ⊢ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))
4	2, 3	2th 267	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
5	4	df-sb 2102	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	2, 5	mpbir 234	1 ⊢ [𝑡 / 𝑥]𝜑

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  [wsb 2101

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828

This proof depends on definitions:  df-bi 210  df-sb 2102

This theorem is used by:  sbtru  2106  sbimi  2113  vexw  2747  iscatd2  17905  iuninc  33253  suppss2f  33330  esumpfinvalf  34819  sbtT  45810  2sb5ndVD  46152  2sb5ndALT  46174  icht  48807