Theorem rp-fakeanorass 37858

Description: A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)

Assertion

Ref	Expression
rp-fakeanorass	⊢ ((𝜒 → 𝜑) ↔ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ (𝜑 ∧ (𝜓 ∨ 𝜒))))

Proof of Theorem rp-fakeanorass

Step	Hyp	Ref	Expression
1		pm1.4 401	. . . . . . . 8 ⊢ ((𝜑 ∨ 𝜒) → (𝜒 ∨ 𝜑))
2	1	ord 392	. . . . . . 7 ⊢ ((𝜑 ∨ 𝜒) → (¬ 𝜒 → 𝜑))
3		pm4.83 970	. . . . . . . 8 ⊢ (((𝜒 → 𝜑) ∧ (¬ 𝜒 → 𝜑)) ↔ 𝜑)
4	3	biimpi 206	. . . . . . 7 ⊢ (((𝜒 → 𝜑) ∧ (¬ 𝜒 → 𝜑)) → 𝜑)
5	2, 4	sylan2 491	. . . . . 6 ⊢ (((𝜒 → 𝜑) ∧ (𝜑 ∨ 𝜒)) → 𝜑)
6	5	ex 450	. . . . 5 ⊢ ((𝜒 → 𝜑) → ((𝜑 ∨ 𝜒) → 𝜑))
7	6	anim1d 588	. . . 4 ⊢ ((𝜒 → 𝜑) → (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒))))
8		orc 400	. . . . 5 ⊢ (𝜑 → (𝜑 ∨ 𝜒))
9	8	anim1i 592	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))
10	7, 9	jctir 561	. . 3 ⊢ ((𝜒 → 𝜑) → ((((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒))) ∧ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))))
11		olc 399	. . . . . 6 ⊢ (𝜒 → (𝜑 ∨ 𝜒))
12		olc 399	. . . . . 6 ⊢ (𝜒 → (𝜓 ∨ 𝜒))
13	11, 12	jca 554	. . . . 5 ⊢ (𝜒 → ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))
14		simpl 473	. . . . 5 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → 𝜑)
15	13, 14	imim12i 62	. . . 4 ⊢ ((((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒))) → (𝜒 → 𝜑))
16	15	adantr 481	. . 3 ⊢ (((((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒))) ∧ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))) → (𝜒 → 𝜑))
17	10, 16	impbii 199	. 2 ⊢ ((𝜒 → 𝜑) ↔ ((((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒))) ∧ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))))
18		dfbi2 660	. 2 ⊢ ((((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ (𝜑 ∧ (𝜓 ∨ 𝜒))) ↔ ((((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒))) ∧ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))))
19		ordir 909	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))
20	19	bicomi 214	. . 3 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ 𝜒))
21	20	bibi1i 328	. 2 ⊢ ((((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ (𝜑 ∧ (𝜓 ∨ 𝜒))) ↔ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ (𝜑 ∧ (𝜓 ∨ 𝜒))))
22	17, 18, 21	3bitr2i 288	1 ⊢ ((𝜒 → 𝜑) ↔ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ (𝜑 ∧ (𝜓 ∨ 𝜒))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  rp-fakeoranass  37859  rp-fakeinunass  37861