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Theorem sbbi 2388
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 657 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21sbbii 1873 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)))
3 sbim 2382 . . . 4 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
4 sbim 2382 . . . 4 ([𝑦 / 𝑥](𝜓𝜑) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑))
53, 4anbi12i 728 . . 3 (([𝑦 / 𝑥](𝜑𝜓) ∧ [𝑦 / 𝑥](𝜓𝜑)) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
6 sban 2386 . . 3 ([𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ([𝑦 / 𝑥](𝜑𝜓) ∧ [𝑦 / 𝑥](𝜓𝜑)))
7 dfbi2 657 . . 3 (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
85, 6, 73bitr4i 290 . 2 ([𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
92, 8bitri 262 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  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 2032  ax-13 2232
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:  spsbbi  2389  sblbis  2391  sbrbis  2392  pm13.183  3312  sbcbig  3446  sb8iota  5761  bj-sbieOLD  31833  bj-sbidmOLD  31834
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