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Theorem sblim 2068
Description: Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sblim.1 xψ
Assertion
Ref Expression
sblim ([y / x](φψ) ↔ ([y / x]φψ))

Proof of Theorem sblim
StepHypRef Expression
1 sbim 2065 . 2 ([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))
2 sblim.1 . . . 4 xψ
32sbf 2026 . . 3 ([y / x]ψψ)
43imbi2i 303 . 2 (([y / x]φ → [y / x]ψ) ↔ ([y / x]φψ))
51, 4bitri 240 1 ([y / x](φψ) ↔ ([y / x]φψ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176  wnf 1544  [wsb 1648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  sbnf2  2108  sbmo  2234
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