Theorem sbt 2067

Description: A substitution into a theorem yields a theorem. See sbtALT 2070 for a shorter proof requiring more axioms. See chvar 2394 and chvarv 2395 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 2066. (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.

Step	Hyp	Ref	Expression
1		df-sb 2066	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		sbt.1	. . . . 5 ⊢ 𝜑
3	2	a1i 11	. . . 4 ⊢ (𝑥 = 𝑦 → 𝜑)
4	3	ax-gen 1795	. . 3 ⊢ ∀𝑥(𝑥 = 𝑦 → 𝜑)
5	4	a1i 11	. 2 ⊢ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6	1, 5	mpgbir 1799	1 ⊢ [𝑡 / 𝑥]𝜑

Syntax hints:   → wi 4  ∀wal 1538  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795

This theorem depends on definitions:  df-bi 207  df-sb 2066

This theorem is referenced by:  sbtru  2068  sbimi  2075  vexw  2714  iscatd2  17649  iuninc  32496  suppss2f  32569  esumpfinvalf  34073  sbtT  44564  2sb5ndVD  44906  2sb5ndALT  44928  icht  47457