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Theorem sbbi 2389
 Description: Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
sbbi ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))

Proof of Theorem sbbi
StepHypRef Expression
1 dfbi2 658 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21sbbii 1874 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)))
3 sbim 2383 . . . 4 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
4 sbim 2383 . . . 4 ([𝑦 / 𝑥](𝜓𝜑) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑))
53, 4anbi12i 729 . . 3 (([𝑦 / 𝑥](𝜑𝜓) ∧ [𝑦 / 𝑥](𝜓𝜑)) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
6 sban 2387 . . 3 ([𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ([𝑦 / 𝑥](𝜑𝜓) ∧ [𝑦 / 𝑥](𝜓𝜑)))
7 dfbi2 658 . . 3 (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
85, 6, 73bitr4i 291 . 2 ([𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
92, 8bitri 263 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  spsbbi  2390  sblbis  2392  sbrbis  2393  pm13.183  3313  sbcbig  3447  sb8iota  5775  bj-sbieOLD  32020  bj-sbidmOLD  32021
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