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Theorem sbim 2065
 Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbim ([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))

Proof of Theorem sbim
StepHypRef Expression
1 sbi1 2063 . 2 ([y / x](φψ) → ([y / x]φ → [y / x]ψ))
2 sbi2 2064 . 2 (([y / x]φ → [y / x]ψ) → [y / x](φψ))
31, 2impbii 180 1 ([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sbor  2066  sbrim  2067  sblim  2068  sban  2069  sbbi  2071  sbequ8  2079  sbcimg  3087
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