Theorem sbt 1777

Description: A substitution into a theorem remains true. (See chvar 1750 and chvarv 1930 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
sbt.1	⊢ 𝜑

Assertion

Ref	Expression
sbt	⊢ [𝑦 / 𝑥]𝜑

Proof of Theorem sbt

Step	Hyp	Ref	Expression
1		sbt.1	. 2 ⊢ 𝜑
2	1	nfth 1457	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	sbf 1770	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	1, 3	mpbir 145	1 ⊢ [𝑦 / 𝑥]𝜑

Syntax hints:  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  vjust  2731