Theorem wloglei 11716

Description: Form of wlogle 11717 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 484	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑆 ⊆ ℝ)
3		simprr 782	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
4	2, 3	sseldd 3937	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ ℝ)
5		simprl 780	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
6	2, 5	sseldd 3937	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ ℝ)
7		vex 3457	. . 3 ⊢ 𝑥 ∈ V
8		vex 3457	. . 3 ⊢ 𝑦 ∈ V
9		eleq1w 2844	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆))
10		eleq1w 2844	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
11	9, 10	bi2anan9 647	. . . . . 6 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)))
12	11	anbi2d 639	. . . . 5 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))))
13		breq12 5104	. . . . . 6 ⊢ ((𝑤 = 𝑦 ∧ 𝑧 = 𝑥) → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
14	13	ancoms 462	. . . . 5 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
15	12, 14	anbi12d 641	. . . 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 346	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓) ↔ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑦 ≤ 𝑥) → 𝜒)))
18		vex 3457	. . . 4 ⊢ 𝑧 ∈ V
19		vex 3457	. . . 4 ⊢ 𝑤 ∈ V
20		ancom 464	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ↔ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆))
21		eleq1w 2844	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑆 ↔ 𝑧 ∈ 𝑆))
22		eleq1w 2844	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝑆 ↔ 𝑤 ∈ 𝑆))
23	21, 22	bi2anan9 647	. . . . . . . 8 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) ↔ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)))
24	20, 23	bitrid 285	. . . . . . 7 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ↔ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)))
25	24	anbi2d 639	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ↔ (𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))))
26		breq12 5104	. . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧))
27	26	ancoms 462	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧))
28	25, 27	anbi12d 641	. . . . 5 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) ↔ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧)))
29		equcom 2037	. . . . . . 7 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
30		equcom 2037	. . . . . . 7 ⊢ (𝑥 = 𝑤 ↔ 𝑤 = 𝑥)
31		wlogle.2	. . . . . . 7 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃))
32	29, 30, 31	syl2anb 607	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝜓 ↔ 𝜃))
33	32	bicomd 225	. . . . 5 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝜃 ↔ 𝜓))
34	28, 33	imbi12d 346	. . . 4 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜃) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓)))
35		df-3an 1099	. . . . . 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 604	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ≤ 𝑦)) → 𝜃)
38	37	anassrs 471	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜃)
39	18, 19, 34, 38	vtocl2 3531	. . 3 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ 𝑤 ≤ 𝑧) → 𝜓)
40	7, 8, 17, 39	vtocl2 3531	. 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 604	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ≤ 𝑦)) → 𝜒)
43	42	anassrs 471	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑥 ≤ 𝑦) → 𝜒)
44	4, 6, 40, 43	lecasei 11286	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   ∈ wcel 2141   ⊆ wss 3904   class class class wbr 5099  ℝcr 11069   ≤ cle 11214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-resscn 11127  ax-pre-lttri 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219

This theorem is referenced by:  wlogle  11717  resconn  35560