Theorem eupth2lem2dc 16614

Description: Lemma for eupth2 . (Contributed by Mario Carneiro, 8-Apr-2015.)

Hypotheses

Ref	Expression
eupth2lem2dc.1	⊢ (𝜑 → 𝐵 ∈ 𝑋)
eupth2lem2dc.dc	⊢ (𝜑 → DECID 𝐴 = 𝐵)
eupth2lem2dc.bc	⊢ (𝜑 → 𝐵 ≠ 𝐶)
eupth2lem2dc.bu	⊢ (𝜑 → 𝐵 = 𝑈)

Assertion

Ref	Expression
eupth2lem2dc	⊢ (𝜑 → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))

Proof of Theorem eupth2lem2dc

Step	Hyp	Ref	Expression
1		eupth2lem2dc.dc	. . 3 ⊢ (𝜑 → DECID 𝐴 = 𝐵)
2		eqidd 2239	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = 𝐵)
3	2	olcd 746	. . . . . . 7 ⊢ (𝜑 → (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))
4	3	biantrud 304	. . . . . 6 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ (𝐴 ≠ 𝐵 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))))
5		eupth2lem2dc.1	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
6		eupth2lem1 16613	. . . . . . 7 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))))
7	5, 6	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))))
8		eupth2lem2dc.bu	. . . . . . 7 ⊢ (𝜑 → 𝐵 = 𝑈)
9	8	eleq1d 2307	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
10	4, 7, 9	3bitr2d 216	. . . . 5 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
11	10	a1d 22	. . . 4 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}))))
12	11	necon1bbiddc 2483	. . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝐴 = 𝐵)))
13	1, 12	mpd 13	. 2 ⊢ (𝜑 → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝐴 = 𝐵))
14		eupth2lem2dc.bc	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
15		neeq1 2433	. . . . . . 7 ⊢ (𝐵 = 𝐴 → (𝐵 ≠ 𝐶 ↔ 𝐴 ≠ 𝐶))
16	14, 15	syl5ibcom 155	. . . . . 6 ⊢ (𝜑 → (𝐵 = 𝐴 → 𝐴 ≠ 𝐶))
17	16	pm4.71rd 398	. . . . 5 ⊢ (𝜑 → (𝐵 = 𝐴 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 = 𝐴)))
18		eqcom 2240	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
19		ancom 266	. . . . 5 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ≠ 𝐶) ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 = 𝐴))
20	17, 18, 19	3bitr4g 223	. . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐵 = 𝐴 ∧ 𝐴 ≠ 𝐶)))
21	14	neneqd 2441	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐵 = 𝐶)
22		biorf 756	. . . . . . 7 ⊢ (¬ 𝐵 = 𝐶 → (𝐵 = 𝐴 ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐴)))
23	21, 22	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐵 = 𝐴 ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐴)))
24		orcom 740	. . . . . 6 ⊢ ((𝐵 = 𝐶 ∨ 𝐵 = 𝐴) ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))
25	23, 24	bitrdi 196	. . . . 5 ⊢ (𝜑 → (𝐵 = 𝐴 ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
26	25	anbi1d 469	. . . 4 ⊢ (𝜑 → ((𝐵 = 𝐴 ∧ 𝐴 ≠ 𝐶) ↔ ((𝐵 = 𝐴 ∨ 𝐵 = 𝐶) ∧ 𝐴 ≠ 𝐶)))
27	20, 26	bitrd 188	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ((𝐵 = 𝐴 ∨ 𝐵 = 𝐶) ∧ 𝐴 ≠ 𝐶)))
28	27	biancomd 271	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))))
29		eupth2lem1 16613	. . . 4 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ (𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))))
30	5, 29	syl 14	. . 3 ⊢ (𝜑 → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ (𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))))
31	8	eleq1d 2307	. . 3 ⊢ (𝜑 → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
32	30, 31	bitr3d 190	. 2 ⊢ (𝜑 → ((𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
33	13, 28, 32	3bitrd 214	1 ⊢ (𝜑 → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∅c0 3520  ifcif 3635  {cpr 3706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-if 3636  df-sn 3711  df-pr 3712

This theorem is referenced by:  eupth2lem3lem4fi  16628