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Theorem sbbi 2548
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 460 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21sbbii 2056 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)))
3 sbim 2542 . . . 4 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
4 sbim 2542 . . . 4 ([𝑦 / 𝑥](𝜓𝜑) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑))
53, 4anbi12i 612 . . 3 (([𝑦 / 𝑥](𝜑𝜓) ∧ [𝑦 / 𝑥](𝜓𝜑)) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
6 sban 2546 . . 3 ([𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ([𝑦 / 𝑥](𝜑𝜓) ∧ [𝑦 / 𝑥](𝜓𝜑)))
7 dfbi2 460 . . 3 (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
85, 6, 73bitr4i 292 . 2 ([𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
92, 8bitri 264 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  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 837  df-ex 1853  df-nf 1858  df-sb 2050
This theorem is referenced by:  spsbbi  2549  sblbis  2551  sbrbis  2552  pm13.183  3495  sbcbig  3632  sb8iota  6001  bj-sbidmOLD  33166
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