Theorem sbt 1709

Description: A substitution into a theorem remains true. (See chvar 1682 and chvarv 1855 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 1394	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	sbf 1702	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	1, 3	mpbir 144	1 ⊢ [𝑦 / 𝑥]𝜑

Syntax hints:  [wsb 1687

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688

This theorem is referenced by:  vjust  2611