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Theorem biadan 818
Description: An implication is equivalent to the equivalence of some implied equivalence and some other equivalence involving a conjunction. A utility lemma as illustrated in biadanii 821 and elelb 34337. (Contributed by BJ, 4-Mar-2023.) (Proof shortened by Wolf Lammen, 8-Mar-2023.)
Assertion
Ref Expression
biadan ((𝜑𝜓) ↔ ((𝜓 → (𝜑𝜒)) ↔ (𝜑 ↔ (𝜓𝜒))))

Proof of Theorem biadan
StepHypRef Expression
1 pm4.71r 562 . 2 ((𝜑𝜓) ↔ (𝜑 ↔ (𝜓𝜑)))
2 bicom 225 . 2 ((𝜑 ↔ (𝜓𝜑)) ↔ ((𝜓𝜑) ↔ 𝜑))
3 bicom 225 . . . 4 ((𝜑 ↔ (𝜓𝜒)) ↔ ((𝜓𝜒) ↔ 𝜑))
4 pm5.32 577 . . . 4 ((𝜓 → (𝜑𝜒)) ↔ ((𝜓𝜑) ↔ (𝜓𝜒)))
53, 4bibi12i 343 . . 3 (((𝜑 ↔ (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒))) ↔ (((𝜓𝜒) ↔ 𝜑) ↔ ((𝜓𝜑) ↔ (𝜓𝜒))))
6 bicom 225 . . 3 (((𝜓 → (𝜑𝜒)) ↔ (𝜑 ↔ (𝜓𝜒))) ↔ ((𝜑 ↔ (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒))))
7 biluk 390 . . 3 (((𝜓𝜑) ↔ 𝜑) ↔ (((𝜓𝜒) ↔ 𝜑) ↔ ((𝜓𝜑) ↔ (𝜓𝜒))))
85, 6, 73bitr4ri 307 . 2 (((𝜓𝜑) ↔ 𝜑) ↔ ((𝜓 → (𝜑𝜒)) ↔ (𝜑 ↔ (𝜓𝜒))))
91, 2, 83bitri 300 1 ((𝜑𝜓) ↔ ((𝜓 → (𝜑𝜒)) ↔ (𝜑 ↔ (𝜓𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 210  df-an 400
This theorem is referenced by:  biadani  819
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