Theorem sbim 2336

Description: Implication inside and outside of a 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 2102	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sbi2 2335	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
3	1, 2	impbii 211	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 208  [wsb 2089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-nf 1803  df-sb 2090

This theorem is referenced by:  sblim  2338  sbor  2339  sbbi  2340  sbnf  2344  sbequ8  2531  sbcimg  3792  mo5f  32636  iuninc  32709  suppss2f  32790  esumpfinvalf  34334  wl-sbrimt  38014  wl-sblimt  38015  frege58bcor  44443  frege60b  44445  frege65b  44450  ellimcabssub0  46157