Theorem sbtT 45336

Description: A substitution into a theorem remains true. sbt 2103 with the existence of no virtual hypotheses for the hypothesis expressed as the empty virtual hypothesis collection. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sbtT.1	⊢ (⊤ → 𝜑)

Assertion

Ref	Expression
sbtT	⊢ [𝑦 / 𝑥]𝜑

Proof of Theorem sbtT

Step	Hyp	Ref	Expression
1		sbtT.1	. . 3 ⊢ (⊤ → 𝜑)
2	1	mptru 1577	. 2 ⊢ 𝜑
3	2	sbt 2103	1 ⊢ [𝑦 / 𝑥]𝜑

This proof depends on syntax axioms:   → wi 4  ⊤wtru 1571  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828

This proof depends on definitions:  df-bi 210  df-tru 1573  df-sb 2100

This theorem is used by: (None)