Theorem sbimi 2075

Description: Distribute substitution over implication. (Contributed by NM, 25-Jun-1998.) Revise df-sb 2066. (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 2067	. 2 ⊢ [𝑡 / 𝑥](𝜑 → 𝜓)
3		sbi1 2072	. 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 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-sb 2066

This theorem is referenced by:  sb2imi  2076  sbbii  2077  sban  2081  hbsbw  2172  sb4av  2245  sbi2  2302  hbsb3  2485  sb6f  2495  sbie  2500  2mo  2641  sbhypf  3501  elrabi  3645  fmptdF  32613  funcnv4mpt  32626  disjdsct  32659  measiuns  34183  ballotlemodife  34465  subsym1  36400  bj-hbsb3v  36788  bj-sbidmOLD  36823  mptsnunlem  37311  sbor2  42185