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Theorem wlogle 11689
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 1172 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → (𝜒𝜃))
74, 6mpbid 231 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜃)
81, 2, 3, 7, 4wloglei 11688 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wcel 2107  wss 3911   class class class wbr 5106  cr 11051  cle 11191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-resscn 11109  ax-pre-lttri 11126
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196
This theorem is referenced by:  vdwlem12  16865  iundisj2  24916  volcn  24973  dvlip  25360  ftc1a  25404  iundisj2f  31511  iundisj2fi  31703  erdszelem9  33796  ftc1anc  36162
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