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Theorem sban 2386
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 2378 . . 3 ([𝑦 / 𝑥] ¬ (𝜑 → ¬ 𝜓) ↔ ¬ [𝑦 / 𝑥](𝜑 → ¬ 𝜓))
2 sbim 2382 . . . 4 ([𝑦 / 𝑥](𝜑 → ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥] ¬ 𝜓))
3 sbn 2378 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑦 / 𝑥]𝜓)
43imbi2i 324 . . . 4 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥] ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓))
52, 4bitri 262 . . 3 ([𝑦 / 𝑥](𝜑 → ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓))
61, 5xchbinx 322 . 2 ([𝑦 / 𝑥] ¬ (𝜑 → ¬ 𝜓) ↔ ¬ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓))
7 df-an 384 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
87sbbii 1873 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥] ¬ (𝜑 → ¬ 𝜓))
9 df-an 384 . 2 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ¬ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓))
106, 8, 93bitr4i 290 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  [wsb 1866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2033  ax-13 2233
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867
This theorem is referenced by:  sb3an  2387  sbbi  2388  sbabel  2778  cbvreu  3144  sbcan  3444  rmo3  3493  inab  3853  difab  3854  exss  4852  inopab  5162  mo5f  28514  rmo3f  28525  iuninc  28567  suppss2f  28625  fmptdF  28642  disjdsct  28669  esumpfinvalf  29271  measiuns  29413  ballotlemodife  29692  sb5ALT  37548  sbcangOLD  37556  2uasbanh  37594  2uasbanhVD  37965  sb5ALTVD  37967  ellimcabssub0  38481
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