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Theorem wemaplem3 7520
 Description: Lemma for wemapso 7523. 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 7518 . . . 4
62, 3, 5syl2anc 644 . . 3
71, 6mpbid 203 . 2
8 wemaplem3.xq . . 3
9 wemaplem2.q . . . 4
104wemaplem1 7518 . . . 4
113, 9, 10syl2anc 644 . . 3
128, 11mpbid 203 . 2
13 wemaplem2.a . . . . . 6
1413ad2antrr 708 . . . . 5
152ad2antrr 708 . . . . 5
163ad2antrr 708 . . . . 5
179ad2antrr 708 . . . . 5
18 wemaplem2.r . . . . . 6
1918ad2antrr 708 . . . . 5
20 wemaplem2.s . . . . . 6
2120ad2antrr 708 . . . . 5
22 simplrl 738 . . . . 5
23 simp2rl 1027 . . . . . 6
24233expa 1154 . . . . 5
25 simprr 735 . . . . . 6
2625ad2antlr 709 . . . . 5
27 simprl 734 . . . . 5
28 simprrl 742 . . . . 5
29 simprrr 743 . . . . 5
304, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 28, 29wemaplem2 7519 . . . 4
3130rexlimdvaa 2833 . . 3
3231rexlimdvaa 2833 . 2
337, 12, 32mp2d 44 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  wrex 2708  cvv 2958   class class class wbr 4215  copab 4268   wpo 4504   wor 4505  cfv 5457  (class class class)co 6084   cmap 7021 This theorem is referenced by:  wemappo  7521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-map 7023
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