Theorem sbt 2447

Description: A substitution into a theorem yields a theorem. (See chvar 2298 and chvarv 2299 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.)

Hypothesis

Ref	Expression
sbt.1	⊢ 𝜑

Assertion

Ref	Expression
sbt	⊢ [𝑦 / 𝑥]𝜑

Proof of Theorem sbt

Step	Hyp	Ref	Expression
1		stdpc4 2381	. 2 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2		sbt.1	. 2 ⊢ 𝜑
3	1, 2	mpg 1764	1 ⊢ [𝑦 / 𝑥]𝜑

Syntax hints:  [wsb 1937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-sb 1938

This theorem is referenced by:  vjust  3232  iscatd2  16389  iuninc  29505  suppss2f  29567  esumpfinvalf  30266  sbtT  39100  2sb5ndVD  39460  2sb5ndALT  39482