Theorem sbimi 2079

Description: Distribute substitution over implication. (Contributed by NM, 25-Jun-1998.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.)

Hypothesis

Ref	Expression
sbimi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
sbimi	⊢ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)

Proof of Theorem sbimi

Step	Hyp	Ref	Expression
1		sbimi.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	sbt 2071	. 2 ⊢ [𝑡 / 𝑥](𝜑 → 𝜓)
3		sbi1 2076	. 2 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
4	2, 3	ax-mp 5	1 ⊢ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)

Syntax hints:   → wi 4  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 210  df-sb 2070

This theorem is referenced by:  sb2imi  2080  sbbii  2081  sban  2085  sb4av  2242  sbi2  2306  hbsb3  2505  sb6f  2515  sbie  2521  2mo  2710  fmptdF  30419  funcnv4mpt  30432  disjdsct  30462  measiuns  31586  ballotlemodife  31865  bj-hbsb3v  34252  bj-sbidmOLD  34289  mptsnunlem  34755  sbor2  39395