Theorem wloglei 11507

Description: Form of wlogle 11508 where both sides of the equivalence are proven rather than showing that they are equivalent to each other. (Contributed by Mario Carneiro, 9-Mar-2015.)

Hypotheses

Ref	Expression
wlogle.1	⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝜓 ↔ 𝜒))
wlogle.2	⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃))
wlogle.3	⊢ (𝜑 → 𝑆 ⊆ ℝ)
wloglei.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜃)
wloglei.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜒)

Assertion

Ref	Expression
wloglei	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝜑   𝑤,𝑆,𝑥,𝑦,𝑧   𝜓,𝑥,𝑦   𝜒,𝑤,𝑧

Allowed substitution hints:   𝜓(𝑧,𝑤)   𝜒(𝑥,𝑦)   𝜃(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wloglei

Step	Hyp	Ref	Expression
1		wlogle.3	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ ℝ)
2	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑆 ⊆ ℝ)
3		simprr 770	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
4	2, 3	sseldd 3922	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ ℝ)
5		simprl 768	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
6	2, 5	sseldd 3922	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ ℝ)
7		vex 3436	. . 3 ⊢ 𝑥 ∈ V
8		vex 3436	. . 3 ⊢ 𝑦 ∈ V
9		eleq1w 2821	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆))
10		eleq1w 2821	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
11	9, 10	bi2anan9 636	. . . . . 6 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)))
12	11	anbi2d 629	. . . . 5 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))))
13		breq12 5079	. . . . . 6 ⊢ ((𝑤 = 𝑦 ∧ 𝑧 = 𝑥) → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
14	13	ancoms 459	. . . . 5 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
15	12, 14	anbi12d 631	. . . 4 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) ↔ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑦 ≤ 𝑥)))
16		wlogle.1	. . . 4 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝜓 ↔ 𝜒))
17	15, 16	imbi12d 345	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓) ↔ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑦 ≤ 𝑥) → 𝜒)))
18		vex 3436	. . . 4 ⊢ 𝑧 ∈ V
19		vex 3436	. . . 4 ⊢ 𝑤 ∈ V
20		ancom 461	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ↔ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆))
21		eleq1w 2821	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑆 ↔ 𝑧 ∈ 𝑆))
22		eleq1w 2821	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝑆 ↔ 𝑤 ∈ 𝑆))
23	21, 22	bi2anan9 636	. . . . . . . 8 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) ↔ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)))
24	20, 23	bitrid 282	. . . . . . 7 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ↔ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)))
25	24	anbi2d 629	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ↔ (𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))))
26		breq12 5079	. . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧))
27	26	ancoms 459	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧))
28	25, 27	anbi12d 631	. . . . 5 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) ↔ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧)))
29		equcom 2021	. . . . . . 7 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
30		equcom 2021	. . . . . . 7 ⊢ (𝑥 = 𝑤 ↔ 𝑤 = 𝑥)
31		wlogle.2	. . . . . . 7 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃))
32	29, 30, 31	syl2anb 598	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝜓 ↔ 𝜃))
33	32	bicomd 222	. . . . 5 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝜃 ↔ 𝜓))
34	28, 33	imbi12d 345	. . . 4 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜃) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓)))
35		df-3an 1088	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦) ↔ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ≤ 𝑦))
36		wloglei.4	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜃)
37	35, 36	sylan2br 595	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ≤ 𝑦)) → 𝜃)
38	37	anassrs 468	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜃)
39	18, 19, 34, 38	vtocl2 3500	. . 3 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓)
40	7, 8, 17, 39	vtocl2 3500	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑦 ≤ 𝑥) → 𝜒)
41		wloglei.5	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜒)
42	35, 41	sylan2br 595	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ≤ 𝑦)) → 𝜒)
43	42	anassrs 468	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜒)
44	4, 6, 40, 43	lecasei 11081	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   ∈ wcel 2106   ⊆ wss 3887   class class class wbr 5074  ℝcr 10870   ≤ cle 11010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-resscn 10928  ax-pre-lttri 10945

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015

This theorem is referenced by:  wlogle  11508  resconn  33208