Theorem sbim 2302

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 2070	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sbi2 2301	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
3	1, 2	impbii 209	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 206  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-sb 2064

This theorem is referenced by:  sbrimOLD  2304  sblim  2305  sbor  2306  sbbi  2307  sbnf  2311  sbnfOLD  2312  sbequ8  2504  sbcimg  3819  mo5f  32437  iuninc  32509  suppss2f  32584  esumpfinvalf  34052  wl-sbrimt  37523  wl-sblimt  37524  frege58bcor  43893  frege60b  43895  frege65b  43900  ellimcabssub0  45604