Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbbivOLD Structured version   Visualization version   GIF version

Theorem sbbivOLD 2322
 Description: Obsolete version of sbbi 2315 as of 24-Jul-2023. Substitution distributes over a biconditional. Version of sbbi 2315 with a disjoint variable condition, not requiring ax-11 2159 nor ax-13 2382. (Contributed by Wolf Lammen, 18-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbbivOLD ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbbivOLD
StepHypRef Expression
1 dfbi2 478 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21sbbii 2081 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)))
3 sbimvOLD 2320 . . . 4 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
4 sbimvOLD 2320 . . . 4 ([𝑦 / 𝑥](𝜓𝜑) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑))
53, 4anbi12i 629 . . 3 (([𝑦 / 𝑥](𝜑𝜓) ∧ [𝑦 / 𝑥](𝜓𝜑)) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
6 sbanvOLD 2321 . . 3 ([𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ([𝑦 / 𝑥](𝜑𝜓) ∧ [𝑦 / 𝑥](𝜓𝜑)))
7 dfbi2 478 . . 3 (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
85, 6, 73bitr4i 306 . 2 ([𝑦 / 𝑥]((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
92, 8bitri 278 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by:  sblbisvOLD  2323
 Copyright terms: Public domain W3C validator