Theorem sbt

Description: A substitution into a theorem yields a theorem. See sbtALT for a shorter proof requiring more axioms. See chvar and chvarv 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 . (Revised by Steven Nguyen, 6-Jul-2023)

	Ref	Expression
Hypothesis	sbt.1	|- ph
Assertion	sbt	|- [ t / x ] ph

Proof

Step	Hyp	Ref	Expression
1		sbt.1	|- ph
2	1	a1i	|- ( x = y -> ph )
3	2	ax-gen	|- A. x ( x = y -> ph )
4	3	a1i	|- ( y = t -> A. x ( x = y -> ph ) )
5	4	ax-gen	|- A. y ( y = t -> A. x ( x = y -> ph ) )
6		df-sb	|- ( [ t / x ] ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
7	5 6	mpbir	|- [ t / x ] ph