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Theorem sban 2546
Description: Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
sban ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

Proof of Theorem sban
StepHypRef Expression
1 sbn 2538 . . 3 ([𝑦 / 𝑥] ¬ (𝜑 → ¬ 𝜓) ↔ ¬ [𝑦 / 𝑥](𝜑 → ¬ 𝜓))
2 sbim 2542 . . . 4 ([𝑦 / 𝑥](𝜑 → ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥] ¬ 𝜓))
3 sbn 2538 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑦 / 𝑥]𝜓)
43imbi2i 325 . . . 4 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥] ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓))
52, 4bitri 264 . . 3 ([𝑦 / 𝑥](𝜑 → ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓))
61, 5xchbinx 323 . 2 ([𝑦 / 𝑥] ¬ (𝜑 → ¬ 𝜓) ↔ ¬ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓))
7 df-an 383 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
87sbbii 2056 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥] ¬ (𝜑 → ¬ 𝜓))
9 df-an 383 . 2 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ¬ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓))
106, 8, 93bitr4i 292 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  [wsb 2049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-ex 1853  df-nf 1858  df-sb 2050
This theorem is referenced by:  sb3an  2547  sbbi  2548  sbabel  2942  cbvreu  3318  sbcan  3631  rmo3  3678  inab  4044  difab  4045  exss  5060  inopab  5392  mo5f  29665  rmo3f  29675  iuninc  29718  suppss2f  29780  fmptdF  29797  disjdsct  29821  esumpfinvalf  30479  measiuns  30621  ballotlemodife  30900  sb5ALT  39257  sbcangOLD  39265  2uasbanh  39303  2uasbanhVD  39670  sb5ALTVD  39672  ellimcabssub0  40368
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