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  2303  hbsb3  2492  sb6f  2502  sbie  2507  2mo  2648  sbhypf  3528  elrabi  3671  fmptdF  32639  funcnv4mpt  32652  disjdsct  32685  measiuns  34253  ballotlemodife  34535  subsym1  36450  bj-hbsb3v  36838  bj-sbidmOLD  36873  mptsnunlem  37361  sbor2  42230