Theorem sbt 2033

Description: A substitution into a theorem remains true. (See chvar 1986 and chvarv 2013 for versions using implicit substitution.) (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 1553	. . 3 ⊢ Ⅎxφ
3	2	sbf 2026	. 2 ⊢ ([y / x]φ ↔ φ)
4	1, 3	mpbir 200	1 ⊢ [y / x]φ

Syntax hints:  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  vjust  2861