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Theorem wloglei 10511
Description: Form of wlogle 10512 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 481 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑆 ⊆ ℝ)
3 simprr 795 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝑆)
42, 3sseldd 3588 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦 ∈ ℝ)
5 simprl 793 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
62, 5sseldd 3588 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥 ∈ ℝ)
7 vex 3192 . . 3 𝑥 ∈ V
8 vex 3192 . . 3 𝑦 ∈ V
9 eleq1 2686 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑆𝑥𝑆))
10 eleq1 2686 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑆𝑦𝑆))
119, 10bi2anan9 916 . . . . . 6 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝑧𝑆𝑤𝑆) ↔ (𝑥𝑆𝑦𝑆)))
1211anbi2d 739 . . . . 5 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ↔ (𝜑 ∧ (𝑥𝑆𝑦𝑆))))
13 breq12 4623 . . . . . 6 ((𝑤 = 𝑦𝑧 = 𝑥) → (𝑤𝑧𝑦𝑥))
1413ancoms 469 . . . . 5 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝑤𝑧𝑦𝑥))
1512, 14anbi12d 746 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) ↔ ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥)))
16 wlogle.1 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))
1715, 16imbi12d 334 . . 3 ((𝑧 = 𝑥𝑤 = 𝑦) → ((((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓) ↔ (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥) → 𝜒)))
18 vex 3192 . . . 4 𝑧 ∈ V
19 vex 3192 . . . 4 𝑤 ∈ V
20 ancom 466 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) ↔ (𝑦𝑆𝑥𝑆))
21 eleq1 2686 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦𝑆𝑧𝑆))
22 eleq1 2686 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥𝑆𝑤𝑆))
2321, 22bi2anan9 916 . . . . . . . 8 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝑦𝑆𝑥𝑆) ↔ (𝑧𝑆𝑤𝑆)))
2420, 23syl5bb 272 . . . . . . 7 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝑥𝑆𝑦𝑆) ↔ (𝑧𝑆𝑤𝑆)))
2524anbi2d 739 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ↔ (𝜑 ∧ (𝑧𝑆𝑤𝑆))))
26 breq12 4623 . . . . . . 7 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑥𝑦𝑤𝑧))
2726ancoms 469 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝑥𝑦𝑤𝑧))
2825, 27anbi12d 746 . . . . 5 ((𝑦 = 𝑧𝑥 = 𝑤) → (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) ↔ ((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧)))
29 equcom 1942 . . . . . . 7 (𝑦 = 𝑧𝑧 = 𝑦)
30 equcom 1942 . . . . . . 7 (𝑥 = 𝑤𝑤 = 𝑥)
31 wlogle.2 . . . . . . 7 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))
3229, 30, 31syl2anb 496 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝜓𝜃))
3332bicomd 213 . . . . 5 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝜃𝜓))
3428, 33imbi12d 334 . . . 4 ((𝑦 = 𝑧𝑥 = 𝑤) → ((((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜃) ↔ (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓)))
35 df-3an 1038 . . . . . 6 ((𝑥𝑆𝑦𝑆𝑥𝑦) ↔ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦))
36 wloglei.4 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜃)
3735, 36sylan2br 493 . . . . 5 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦)) → 𝜃)
3837anassrs 679 . . . 4 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜃)
3918, 19, 34, 38vtocl2 3250 . . 3 (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓)
407, 8, 17, 39vtocl2 3250 . 2 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥) → 𝜒)
41 wloglei.5 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)
4235, 41sylan2br 493 . . 3 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦)) → 𝜒)
4342anassrs 679 . 2 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜒)
444, 6, 40, 43lecasei 10094 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  wss 3559   class class class wbr 4618  cr 9886  cle 10026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-resscn 9944  ax-pre-lttri 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031
This theorem is referenced by:  wlogle  10512  resconn  30963
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