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Theorem th3qlem2 7003
 Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
th3q.1
th3q.2
th3q.4
Assertion
Ref Expression
th3qlem2
Distinct variable groups:   ,,,,,,,,,   ,,,,,,,,,   ,,,,,,,   ,,,,,,,   , ,,,,,,,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem th3qlem2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 th3q.2 . . 3
2 eqid 2435 . . . . 5
3 breq1 4207 . . . . . . . 8
43anbi1d 686 . . . . . . 7
5 oveq1 6080 . . . . . . . 8
65breq1d 4214 . . . . . . 7
74, 6imbi12d 312 . . . . . 6
87imbi2d 308 . . . . 5
9 breq2 4208 . . . . . . . 8
109anbi1d 686 . . . . . . 7
11 oveq1 6080 . . . . . . . 8
1211breq2d 4216 . . . . . . 7
1310, 12imbi12d 312 . . . . . 6
1413imbi2d 308 . . . . 5
15 breq1 4207 . . . . . . . . . 10
1615anbi2d 685 . . . . . . . . 9
17 oveq2 6081 . . . . . . . . . 10
1817breq1d 4214 . . . . . . . . 9
1916, 18imbi12d 312 . . . . . . . 8
2019imbi2d 308 . . . . . . 7
21 breq2 4208 . . . . . . . . . 10
2221anbi2d 685 . . . . . . . . 9
23 oveq2 6081 . . . . . . . . . 10
2423breq2d 4216 . . . . . . . . 9
2522, 24imbi12d 312 . . . . . . . 8
2625imbi2d 308 . . . . . . 7
27 th3q.4 . . . . . . . 8
2827expcom 425 . . . . . . 7
292, 20, 26, 282optocl 4945 . . . . . 6
3029com12 29 . . . . 5
312, 8, 14, 302optocl 4945 . . . 4
3231imp 419 . . 3
331, 32th3qlem1 7002 . 2
34 opex 4419 . . . . . 6
35 opex 4419 . . . . . 6
36 eceq1 6933 . . . . . . . . 9
3736eqeq2d 2446 . . . . . . . 8
38 eceq1 6933 . . . . . . . . 9
3938eqeq2d 2446 . . . . . . . 8
4037, 39bi2anan9 844 . . . . . . 7
41 oveq12 6082 . . . . . . . . 9
42 eceq1 6933 . . . . . . . . 9
4341, 42syl 16 . . . . . . . 8
4443eqeq2d 2446 . . . . . . 7
4540, 44anbi12d 692 . . . . . 6
4634, 35, 45spc2ev 3036 . . . . 5
4746exlimivv 1645 . . . 4
4847exlimivv 1645 . . 3
4948moimi 2327 . 2
5033, 49syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wex 1550   wceq 1652   wcel 1725  wmo 2281  cvv 2948  cop 3809   class class class wbr 4204   cxp 4868  (class class class)co 6073   wer 6894  cec 6895  cqs 6896 This theorem is referenced by:  th3qcor  7004  th3q  7005 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fv 5454  df-ov 6076  df-er 6897  df-ec 6899  df-qs 6903
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