Theorem wloglei 10560

Description: Form of wlogle 10561 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 796	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
4	2, 3	sseldd 3604	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ ℝ)
5		simprl 794	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
6	2, 5	sseldd 3604	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ ℝ)
7		vex 3203	. . 3 ⊢ 𝑥 ∈ V
8		vex 3203	. . 3 ⊢ 𝑦 ∈ V
9		eleq1 2689	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆))
10		eleq1 2689	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
11	9, 10	bi2anan9 917	. . . . . 6 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)))
12	11	anbi2d 740	. . . . 5 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))))
13		breq12 4658	. . . . . 6 ⊢ ((𝑤 = 𝑦 ∧ 𝑧 = 𝑥) → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
14	13	ancoms 469	. . . . 5 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
15	12, 14	anbi12d 747	. . . 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 334	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓) ↔ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑦 ≤ 𝑥) → 𝜒)))
18		vex 3203	. . . 4 ⊢ 𝑧 ∈ V
19		vex 3203	. . . 4 ⊢ 𝑤 ∈ V
20		ancom 466	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ↔ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆))
21		eleq1 2689	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑆 ↔ 𝑧 ∈ 𝑆))
22		eleq1 2689	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝑆 ↔ 𝑤 ∈ 𝑆))
23	21, 22	bi2anan9 917	. . . . . . . 8 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) ↔ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)))
24	20, 23	syl5bb 272	. . . . . . 7 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ↔ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)))
25	24	anbi2d 740	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ↔ (𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))))
26		breq12 4658	. . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧))
27	26	ancoms 469	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧))
28	25, 27	anbi12d 747	. . . . 5 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) ↔ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧)))
29		equcom 1945	. . . . . . 7 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
30		equcom 1945	. . . . . . 7 ⊢ (𝑥 = 𝑤 ↔ 𝑤 = 𝑥)
31		wlogle.2	. . . . . . 7 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃))
32	29, 30, 31	syl2anb 496	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝜓 ↔ 𝜃))
33	32	bicomd 213	. . . . 5 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝜃 ↔ 𝜓))
34	28, 33	imbi12d 334	. . . 4 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜃) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓)))
35		df-3an 1039	. . . . . 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 493	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ≤ 𝑦)) → 𝜃)
38	37	anassrs 680	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜃)
39	18, 19, 34, 38	vtocl2 3261	. . 3 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓)
40	7, 8, 17, 39	vtocl2 3261	. 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 493	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ≤ 𝑦)) → 𝜒)
43	42	anassrs 680	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜒)
44	4, 6, 40, 43	lecasei 10143	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990   ⊆ wss 3574   class class class wbr 4653  ℝcr 9935   ≤ cle 10075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-pre-lttri 10010

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080

This theorem is referenced by:  wlogle  10561  resconn  31228