Theorem sbtT 37804

Description: A substitution into a theorem remains true. sbt 2407 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	trud 1484	. 2 ⊢ 𝜑
3	2	sbt 2407	1 ⊢ [𝑦 / 𝑥]𝜑

Syntax hints:   → wi 4  ⊤wtru 1476  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868

This theorem is referenced by: (None)