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Theorem sbbi 2071
 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 609 . . 3 ((φψ) ↔ ((φψ) (ψφ)))
21sbbii 1653 . 2 ([y / x](φψ) ↔ [y / x]((φψ) (ψφ)))
3 sbim 2065 . . . 4 ([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))
4 sbim 2065 . . . 4 ([y / x](ψφ) ↔ ([y / x]ψ → [y / x]φ))
53, 4anbi12i 678 . . 3 (([y / x](φψ) [y / x](ψφ)) ↔ (([y / x]φ → [y / x]ψ) ([y / x]ψ → [y / x]φ)))
6 sban 2069 . . 3 ([y / x]((φψ) (ψφ)) ↔ ([y / x](φψ) [y / x](ψφ)))
7 dfbi2 609 . . 3 (([y / x]φ ↔ [y / x]ψ) ↔ (([y / x]φ → [y / x]ψ) ([y / x]ψ → [y / x]φ)))
85, 6, 73bitr4i 268 . 2 ([y / x]((φψ) (ψφ)) ↔ ([y / x]φ ↔ [y / x]ψ))
92, 8bitri 240 1 ([y / x](φψ) ↔ ([y / x]φ ↔ [y / x]ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sblbis  2072  sbrbis  2073  spsbbi  2077  sbco  2083  sbidm  2085  sbal  2127  sb8eu  2222  pm13.183  2979  sbcbig  3092  sb8iota  4346
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