Theorem wloglei 11734

Description: Form of wlogle 11735 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 485	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑆 ⊆ ℝ)
3		simprr 784	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
4	2, 3	sseldd 3940	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ ℝ)
5		simprl 782	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
6	2, 5	sseldd 3940	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ ℝ)
7		vex 3461	. . 3 ⊢ 𝑥 ∈ V
8		vex 3461	. . 3 ⊢ 𝑦 ∈ V
9		eleq1w 2848	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆))
10		eleq1w 2848	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
11	9, 10	bi2anan9 649	. . . . . 6 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)))
12	11	anbi2d 641	. . . . 5 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))))
13		breq12 5110	. . . . . 6 ⊢ ((𝑤 = 𝑦 ∧ 𝑧 = 𝑥) → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
14	13	ancoms 463	. . . . 5 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
15	12, 14	anbi12d 643	. . . 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 347	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓) ↔ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑦 ≤ 𝑥) → 𝜒)))
18		vex 3461	. . . 4 ⊢ 𝑧 ∈ V
19		vex 3461	. . . 4 ⊢ 𝑤 ∈ V
20		ancom 465	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ↔ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆))
21		eleq1w 2848	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑆 ↔ 𝑧 ∈ 𝑆))
22		eleq1w 2848	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝑆 ↔ 𝑤 ∈ 𝑆))
23	21, 22	bi2anan9 649	. . . . . . . 8 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) ↔ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)))
24	20, 23	bitrid 286	. . . . . . 7 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ↔ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)))
25	24	anbi2d 641	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ↔ (𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))))
26		breq12 5110	. . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧))
27	26	ancoms 463	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧))
28	25, 27	anbi12d 643	. . . . 5 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) ↔ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧)))
29		equcom 2041	. . . . . . 7 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
30		equcom 2041	. . . . . . 7 ⊢ (𝑥 = 𝑤 ↔ 𝑤 = 𝑥)
31		wlogle.2	. . . . . . 7 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃))
32	29, 30, 31	syl2anb 609	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝜓 ↔ 𝜃))
33	32	bicomd 226	. . . . 5 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝜃 ↔ 𝜓))
34	28, 33	imbi12d 347	. . . 4 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜃) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓)))
35		df-3an 1103	. . . . . 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 606	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ≤ 𝑦)) → 𝜃)
38	37	anassrs 472	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜃)
39	18, 19, 34, 38	vtocl2 3534	. . 3 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓)
40	7, 8, 17, 39	vtocl2 3534	. 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 606	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ≤ 𝑦)) → 𝜒)
43	42	anassrs 472	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜒)
44	4, 6, 40, 43	lecasei 11304	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   ∈ wcel 2145   ⊆ wss 3907   class class class wbr 5105  ℝcr 11087   ≤ cle 11232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-resscn 11145  ax-pre-lttri 11162

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237

This theorem is referenced by:  wlogle  11735  resconn  35609