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Theorem wlogle 11748
Description: If the predicate 𝜒(𝑥, 𝑦) is symmetric under interchange of 𝑥, 𝑦, then "without loss of generality" we can assume that 𝑥𝑦. (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.)
Hypotheses
Ref Expression
wlogle.1 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))
wlogle.2 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))
wlogle.3 (𝜑𝑆 ⊆ ℝ)
wlogle.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝜒𝜃))
wlogle.5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)
Assertion
Ref Expression
wlogle ((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 wlogle
StepHypRef Expression
1 wlogle.1 . 2 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))
2 wlogle.2 . 2 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))
3 wlogle.3 . 2 (𝜑𝑆 ⊆ ℝ)
4 wlogle.5 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)
5 wlogle.4 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝜒𝜃))
653adantr3 1168 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → (𝜒𝜃))
74, 6mpbid 231 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜃)
81, 2, 3, 7, 4wloglei 11747 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084  wcel 2098  wss 3943   class class class wbr 5141  cr 11108  cle 11250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-resscn 11166  ax-pre-lttri 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255
This theorem is referenced by:  vdwlem12  16932  iundisj2  25429  volcn  25486  dvlip  25877  ftc1a  25923  iundisj2f  32326  iundisj2fi  32513  erdszelem9  34718  ftc1anc  37080
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