Theorem sbim 2383

Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.)

Assertion

Ref	Expression
sbim	⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem sbim

Step	Hyp	Ref	Expression
1		sbi1 2380	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sbi2 2381	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
3	1, 2	impbii 198	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 195  [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-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by:  sbrim  2384  sblim  2385  sbor  2386  sban  2387  sbbi  2389  sbequ8ALT  2395  sbcimg  3444  mo5f  28708  iuninc  28761  suppss2f  28819  esumpfinvalf  29465  bj-sbnf  32016  wl-sbrimt  32510  wl-sblimt  32511  frege58bcor  37217  frege60b  37219  frege65b  37224  ellimcabssub0  38684