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Theorem sbbii 1689
Description: Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
sbbii.1 (𝜑𝜓)
Assertion
Ref Expression
sbbii ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)

Proof of Theorem sbbii
StepHypRef Expression
1 sbbii.1 . . . 4 (𝜑𝜓)
21biimpi 118 . . 3 (𝜑𝜓)
32sbimi 1688 . 2 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)
41biimpri 131 . . 3 (𝜓𝜑)
54sbimi 1688 . 2 ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)
63, 5impbii 124 1 ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)
Colors of variables: wff set class
Syntax hints:  wb 103  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  sbco2v  1863  equsb3  1867  sbn  1868  sbim  1869  sbor  1870  sban  1871  sb3an  1874  sbbi  1875  sbco2h  1880  sbco2d  1882  sbco2vd  1883  sbco3v  1885  sbco3  1890  elsb3  1894  elsb4  1895  sbcom2v2  1904  sbcom2  1905  dfsb7  1909  sb7f  1910  sb7af  1911  sbal  1918  sbal1  1920  sbex  1922  sbco4lem  1924  sbco4  1925  sbmo  2001  eqsb3  2183  clelsb3  2184  clelsb4  2185  sbabel  2245  sbralie  2591  sbcco  2837  exss  3990  inopab  4496  isarep1  5016  bezoutlemnewy  10529
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