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Theorem wloglei 11170
Description: Form of wlogle 11171 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 484 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑆 ⊆ ℝ)
3 simprr 772 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝑆)
42, 3sseldd 3954 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦 ∈ ℝ)
5 simprl 770 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
62, 5sseldd 3954 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥 ∈ ℝ)
7 vex 3483 . . 3 𝑥 ∈ V
8 vex 3483 . . 3 𝑦 ∈ V
9 eleq1w 2898 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑆𝑥𝑆))
10 eleq1w 2898 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑆𝑦𝑆))
119, 10bi2anan9 638 . . . . . 6 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝑧𝑆𝑤𝑆) ↔ (𝑥𝑆𝑦𝑆)))
1211anbi2d 631 . . . . 5 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ↔ (𝜑 ∧ (𝑥𝑆𝑦𝑆))))
13 breq12 5057 . . . . . 6 ((𝑤 = 𝑦𝑧 = 𝑥) → (𝑤𝑧𝑦𝑥))
1413ancoms 462 . . . . 5 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝑤𝑧𝑦𝑥))
1512, 14anbi12d 633 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) ↔ ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥)))
16 wlogle.1 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))
1715, 16imbi12d 348 . . 3 ((𝑧 = 𝑥𝑤 = 𝑦) → ((((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓) ↔ (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥) → 𝜒)))
18 vex 3483 . . . 4 𝑧 ∈ V
19 vex 3483 . . . 4 𝑤 ∈ V
20 ancom 464 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) ↔ (𝑦𝑆𝑥𝑆))
21 eleq1w 2898 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦𝑆𝑧𝑆))
22 eleq1w 2898 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥𝑆𝑤𝑆))
2321, 22bi2anan9 638 . . . . . . . 8 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝑦𝑆𝑥𝑆) ↔ (𝑧𝑆𝑤𝑆)))
2420, 23syl5bb 286 . . . . . . 7 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝑥𝑆𝑦𝑆) ↔ (𝑧𝑆𝑤𝑆)))
2524anbi2d 631 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ↔ (𝜑 ∧ (𝑧𝑆𝑤𝑆))))
26 breq12 5057 . . . . . . 7 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑥𝑦𝑤𝑧))
2726ancoms 462 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝑥𝑦𝑤𝑧))
2825, 27anbi12d 633 . . . . 5 ((𝑦 = 𝑧𝑥 = 𝑤) → (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) ↔ ((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧)))
29 equcom 2026 . . . . . . 7 (𝑦 = 𝑧𝑧 = 𝑦)
30 equcom 2026 . . . . . . 7 (𝑥 = 𝑤𝑤 = 𝑥)
31 wlogle.2 . . . . . . 7 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))
3229, 30, 31syl2anb 600 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝜓𝜃))
3332bicomd 226 . . . . 5 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝜃𝜓))
3428, 33imbi12d 348 . . . 4 ((𝑦 = 𝑧𝑥 = 𝑤) → ((((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜃) ↔ (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓)))
35 df-3an 1086 . . . . . 6 ((𝑥𝑆𝑦𝑆𝑥𝑦) ↔ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦))
36 wloglei.4 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜃)
3735, 36sylan2br 597 . . . . 5 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦)) → 𝜃)
3837anassrs 471 . . . 4 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜃)
3918, 19, 34, 38vtocl2 3547 . . 3 (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓)
407, 8, 17, 39vtocl2 3547 . 2 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥) → 𝜒)
41 wloglei.5 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)
4235, 41sylan2br 597 . . 3 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦)) → 𝜒)
4342anassrs 471 . 2 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜒)
444, 6, 40, 43lecasei 10744 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wcel 2115  wss 3919   class class class wbr 5052  cr 10534  cle 10674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-resscn 10592  ax-pre-lttri 10609
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679
This theorem is referenced by:  wlogle  11171  resconn  32553
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