Theorem sbim 2341

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 2108	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sbi2 2340	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
3	1, 2	impbii 212	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  sblim  2343  sbor  2344  sbbi  2345  sbnf  2349  sbequ8  2536  sbcimg  3795  mo5f  32872  iuninc  32942  suppss2f  33020  esumpfinvalf  34497  wl-sbrimt  38243  wl-sblimt  38244  frege58bcor  44670  frege60b  44672  frege65b  44677  ellimcabssub0  46374