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Theorem sbbi 1849
Description: Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbbi ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))

Proof of Theorem sbbi
StepHypRef Expression
1 dfbi2 374 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21sbbii 1664 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)))
3 sbim 1843 . . . 4 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
4 sbim 1843 . . . 4 ([𝑦 / 𝑥](𝜓𝜑) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑))
53, 4anbi12i 441 . . 3 (([𝑦 / 𝑥](𝜑𝜓) ∧ [𝑦 / 𝑥](𝜓𝜑)) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
6 sban 1845 . . 3 ([𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ([𝑦 / 𝑥](𝜑𝜓) ∧ [𝑦 / 𝑥](𝜓𝜑)))
7 dfbi2 374 . . 3 (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
85, 6, 73bitr4i 205 . 2 ([𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
92, 8bitri 177 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  sblbis  1850  sbrbis  1851  sbco  1858  sbcocom  1860  elsb3  1868  elsb4  1869  sb8eu  1929  sb8euh  1939  pm13.183  2703  sbcbig  2831  sb8iota  4901
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