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Theorem wemaplem3 8072
 Description: Lemma for wemapso 8075. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Hypotheses
Ref Expression
wemapso.t
wemaplem2.a
wemaplem2.p
wemaplem2.x
wemaplem2.q
wemaplem2.r
wemaplem2.s
wemaplem3.px
wemaplem3.xq
Assertion
Ref Expression
wemaplem3
Distinct variable groups:   ,   ,,,,   ,,,,   ,,,,   ,,,,   ,,,,   ,,,,
Allowed substitution hints:   (,,,)   (,,)   (,,,)

Proof of Theorem wemaplem3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wemaplem3.px . . 3
2 wemaplem2.p . . . 4
3 wemaplem2.x . . . 4
4 wemapso.t . . . . 5
54wemaplem1 8070 . . . 4
62, 3, 5syl2anc 665 . . 3
71, 6mpbid 213 . 2
8 wemaplem3.xq . . 3
9 wemaplem2.q . . . 4
104wemaplem1 8070 . . . 4
113, 9, 10syl2anc 665 . . 3
128, 11mpbid 213 . 2
13 wemaplem2.a . . . . . 6
1413ad2antrr 730 . . . . 5
152ad2antrr 730 . . . . 5
163ad2antrr 730 . . . . 5
179ad2antrr 730 . . . . 5
18 wemaplem2.r . . . . . 6
1918ad2antrr 730 . . . . 5
20 wemaplem2.s . . . . . 6
2120ad2antrr 730 . . . . 5
22 simplrl 768 . . . . 5
23 simp2rl 1074 . . . . . 6
24233expa 1205 . . . . 5
25 simprr 764 . . . . . 6
2625ad2antlr 731 . . . . 5
27 simprl 762 . . . . 5
28 simprrl 772 . . . . 5
29 simprrr 773 . . . . 5
304, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 28, 29wemaplem2 8071 . . . 4
3130rexlimdvaa 2915 . . 3
3231rexlimdvaa 2915 . 2
337, 12, 32mp2d 46 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1872  wral 2771  wrex 2772  cvv 3080   class class class wbr 4423  copab 4481   wpo 4772   wor 4773  cfv 5601  (class class class)co 6305   cmap 7483 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7485 This theorem is referenced by:  wemappo  8073
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