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Theorem sblim 1928
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 1924 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sblim.1 . . . 4 𝑥𝜓
32sbf 1750 . . 3 ([𝑦 / 𝑥]𝜓𝜓)
43imbi2i 225 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))
51, 4bitri 183 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1436  [wsb 1735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  sbnf2  1954  sbmo  2056
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