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Theorem wloglei 11751
Description: Form of wlogle 11752 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 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))
wlogle.2 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))
wlogle.3 (𝜑𝑆 ⊆ ℝ)
wloglei.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜃)
wloglei.5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)
Assertion
Ref Expression
wloglei ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 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
StepHypRef Expression
1 wlogle.3 . . . 4 (𝜑𝑆 ⊆ ℝ)
21adantr 480 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑆 ⊆ ℝ)
3 simprr 770 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝑆)
42, 3sseldd 3983 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦 ∈ ℝ)
5 simprl 768 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
62, 5sseldd 3983 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥 ∈ ℝ)
7 vex 3477 . . 3 𝑥 ∈ V
8 vex 3477 . . 3 𝑦 ∈ V
9 eleq1w 2815 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑆𝑥𝑆))
10 eleq1w 2815 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑆𝑦𝑆))
119, 10bi2anan9 636 . . . . . 6 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝑧𝑆𝑤𝑆) ↔ (𝑥𝑆𝑦𝑆)))
1211anbi2d 628 . . . . 5 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ↔ (𝜑 ∧ (𝑥𝑆𝑦𝑆))))
13 breq12 5153 . . . . . 6 ((𝑤 = 𝑦𝑧 = 𝑥) → (𝑤𝑧𝑦𝑥))
1413ancoms 458 . . . . 5 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝑤𝑧𝑦𝑥))
1512, 14anbi12d 630 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) ↔ ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥)))
16 wlogle.1 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))
1715, 16imbi12d 344 . . 3 ((𝑧 = 𝑥𝑤 = 𝑦) → ((((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓) ↔ (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥) → 𝜒)))
18 vex 3477 . . . 4 𝑧 ∈ V
19 vex 3477 . . . 4 𝑤 ∈ V
20 ancom 460 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) ↔ (𝑦𝑆𝑥𝑆))
21 eleq1w 2815 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦𝑆𝑧𝑆))
22 eleq1w 2815 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥𝑆𝑤𝑆))
2321, 22bi2anan9 636 . . . . . . . 8 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝑦𝑆𝑥𝑆) ↔ (𝑧𝑆𝑤𝑆)))
2420, 23bitrid 283 . . . . . . 7 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝑥𝑆𝑦𝑆) ↔ (𝑧𝑆𝑤𝑆)))
2524anbi2d 628 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ↔ (𝜑 ∧ (𝑧𝑆𝑤𝑆))))
26 breq12 5153 . . . . . . 7 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑥𝑦𝑤𝑧))
2726ancoms 458 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝑥𝑦𝑤𝑧))
2825, 27anbi12d 630 . . . . 5 ((𝑦 = 𝑧𝑥 = 𝑤) → (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) ↔ ((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧)))
29 equcom 2020 . . . . . . 7 (𝑦 = 𝑧𝑧 = 𝑦)
30 equcom 2020 . . . . . . 7 (𝑥 = 𝑤𝑤 = 𝑥)
31 wlogle.2 . . . . . . 7 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))
3229, 30, 31syl2anb 597 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝜓𝜃))
3332bicomd 222 . . . . 5 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝜃𝜓))
3428, 33imbi12d 344 . . . 4 ((𝑦 = 𝑧𝑥 = 𝑤) → ((((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜃) ↔ (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓)))
35 df-3an 1088 . . . . . 6 ((𝑥𝑆𝑦𝑆𝑥𝑦) ↔ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦))
36 wloglei.4 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜃)
3735, 36sylan2br 594 . . . . 5 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦)) → 𝜃)
3837anassrs 467 . . . 4 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜃)
3918, 19, 34, 38vtocl2 3554 . . 3 (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓)
407, 8, 17, 39vtocl2 3554 . 2 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥) → 𝜒)
41 wloglei.5 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)
4235, 41sylan2br 594 . . 3 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦)) → 𝜒)
4342anassrs 467 . 2 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜒)
444, 6, 40, 43lecasei 11325 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086  wcel 2105  wss 3948   class class class wbr 5148  cr 11112  cle 11254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-resscn 11170  ax-pre-lttri 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259
This theorem is referenced by:  wlogle  11752  resconn  34536
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