Theorem sbim 2278

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 2275	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sbi2 2276	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
3	1, 2	impbii 194	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 191  [wsb 1828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983  ax-13 2137

This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-ex 1693  df-nf 1697  df-sb 1829

This theorem is referenced by:  sbrim  2279  sblim  2280  sbor  2281  sban  2282  sbbi  2284  sbequ8ALT  2290  sbcimg  3333  mo5f  28281  iuninc  28335  suppss2fOLD  28395  suppss2f  28396  esumpfinvalf  29052  bj-sbnf  31625  wl-sbrimt  32109  wl-sblimt  32110  frege58bcor  36738  frege60b  36740  frege65b  36745  ellimcabssub0  38101