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Theorem wloglei 10772
Description: Form of wlogle 10773 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 472 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑆 ⊆ ℝ)
3 simprr 813 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝑆)
42, 3sseldd 3745 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦 ∈ ℝ)
5 simprl 811 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
62, 5sseldd 3745 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥 ∈ ℝ)
7 vex 3343 . . 3 𝑥 ∈ V
8 vex 3343 . . 3 𝑦 ∈ V
9 eleq1w 2822 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑆𝑥𝑆))
10 eleq1w 2822 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑆𝑦𝑆))
119, 10bi2anan9 953 . . . . . 6 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝑧𝑆𝑤𝑆) ↔ (𝑥𝑆𝑦𝑆)))
1211anbi2d 742 . . . . 5 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ↔ (𝜑 ∧ (𝑥𝑆𝑦𝑆))))
13 breq12 4809 . . . . . 6 ((𝑤 = 𝑦𝑧 = 𝑥) → (𝑤𝑧𝑦𝑥))
1413ancoms 468 . . . . 5 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝑤𝑧𝑦𝑥))
1512, 14anbi12d 749 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) ↔ ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥)))
16 wlogle.1 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))
1715, 16imbi12d 333 . . 3 ((𝑧 = 𝑥𝑤 = 𝑦) → ((((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓) ↔ (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥) → 𝜒)))
18 vex 3343 . . . 4 𝑧 ∈ V
19 vex 3343 . . . 4 𝑤 ∈ V
20 ancom 465 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) ↔ (𝑦𝑆𝑥𝑆))
21 eleq1w 2822 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦𝑆𝑧𝑆))
22 eleq1w 2822 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥𝑆𝑤𝑆))
2321, 22bi2anan9 953 . . . . . . . 8 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝑦𝑆𝑥𝑆) ↔ (𝑧𝑆𝑤𝑆)))
2420, 23syl5bb 272 . . . . . . 7 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝑥𝑆𝑦𝑆) ↔ (𝑧𝑆𝑤𝑆)))
2524anbi2d 742 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ↔ (𝜑 ∧ (𝑧𝑆𝑤𝑆))))
26 breq12 4809 . . . . . . 7 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑥𝑦𝑤𝑧))
2726ancoms 468 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝑥𝑦𝑤𝑧))
2825, 27anbi12d 749 . . . . 5 ((𝑦 = 𝑧𝑥 = 𝑤) → (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) ↔ ((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧)))
29 equcom 2100 . . . . . . 7 (𝑦 = 𝑧𝑧 = 𝑦)
30 equcom 2100 . . . . . . 7 (𝑥 = 𝑤𝑤 = 𝑥)
31 wlogle.2 . . . . . . 7 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))
3229, 30, 31syl2anb 497 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝜓𝜃))
3332bicomd 213 . . . . 5 ((𝑦 = 𝑧𝑥 = 𝑤) → (𝜃𝜓))
3428, 33imbi12d 333 . . . 4 ((𝑦 = 𝑧𝑥 = 𝑤) → ((((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜃) ↔ (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓)))
35 df-3an 1074 . . . . . 6 ((𝑥𝑆𝑦𝑆𝑥𝑦) ↔ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦))
36 wloglei.4 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜃)
3735, 36sylan2br 494 . . . . 5 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦)) → 𝜃)
3837anassrs 683 . . . 4 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜃)
3918, 19, 34, 38vtocl2 3401 . . 3 (((𝜑 ∧ (𝑧𝑆𝑤𝑆)) ∧ 𝑤𝑧) → 𝜓)
407, 8, 17, 39vtocl2 3401 . 2 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑦𝑥) → 𝜒)
41 wloglei.5 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)
4235, 41sylan2br 494 . . 3 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ 𝑥𝑦)) → 𝜒)
4342anassrs 683 . 2 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ 𝑥𝑦) → 𝜒)
444, 6, 40, 43lecasei 10355 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wcel 2139  wss 3715   class class class wbr 4804  cr 10147  cle 10287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-resscn 10205  ax-pre-lttri 10222
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292
This theorem is referenced by:  wlogle  10773  resconn  31556
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