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Theorem th3qlem1 6115
Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
th3qlem1.1 Er 𝑆
th3qlem1.3 (((y 𝑆 w 𝑆) (z 𝑆 v 𝑆)) → ((y w z v) → (y + z) (w + v)))
Assertion
Ref Expression
th3qlem1 ((A (𝑆 / ) B (𝑆 / )) → ∃*xyz((A = [y] B = [z] ) x = [(y + z)] ))
Distinct variable groups:   x,y,z,w,v, +   x, ,y,z,w,v   x,𝑆,y,z,w,v   x,A,y,z,w,v   x,B,y,z,w,v

Proof of Theorem th3qlem1
Dummy variable u is distinct from all other variables.
StepHypRef Expression
1 ee4anv 1787 . . . 4 (yzwv(((A = [y] B = [z] ) x = [(y + z)] ) ((A = [w] B = [v] ) u = [(w + v)] )) ↔ (yz((A = [y] B = [z] ) x = [(y + z)] ) wv((A = [w] B = [v] ) u = [(w + v)] )))
2 an4 507 . . . . . . 7 ((((A = [y] B = [z] ) x = [(y + z)] ) ((A = [w] B = [v] ) u = [(w + v)] )) ↔ (((A = [y] B = [z] ) (A = [w] B = [v] )) (x = [(y + z)] u = [(w + v)] )))
3 eleq1 2078 . . . . . . . . . . . . 13 (A = [y] → (A (𝑆 / ) ↔ [y] (𝑆 / )))
4 eleq1 2078 . . . . . . . . . . . . 13 (B = [z] → (B (𝑆 / ) ↔ [z] (𝑆 / )))
53, 4bi2anan9 526 . . . . . . . . . . . 12 ((A = [y] B = [z] ) → ((A (𝑆 / ) B (𝑆 / )) ↔ ([y] (𝑆 / ) [z] (𝑆 / ))))
65adantr 261 . . . . . . . . . . 11 (((A = [y] B = [z] ) (A = [w] B = [v] )) → ((A (𝑆 / ) B (𝑆 / )) ↔ ([y] (𝑆 / ) [z] (𝑆 / ))))
76biimpac 282 . . . . . . . . . 10 (((A (𝑆 / ) B (𝑆 / )) ((A = [y] B = [z] ) (A = [w] B = [v] ))) → ([y] (𝑆 / ) [z] (𝑆 / )))
8 eqtr2 2036 . . . . . . . . . . . . 13 ((A = [y] A = [w] ) → [y] = [w] )
9 eqtr2 2036 . . . . . . . . . . . . 13 ((B = [z] B = [v] ) → [z] = [v] )
108, 9anim12i 321 . . . . . . . . . . . 12 (((A = [y] A = [w] ) (B = [z] B = [v] )) → ([y] = [w] [z] = [v] ))
1110an4s 509 . . . . . . . . . . 11 (((A = [y] B = [z] ) (A = [w] B = [v] )) → ([y] = [w] [z] = [v] ))
1211adantl 262 . . . . . . . . . 10 (((A (𝑆 / ) B (𝑆 / )) ((A = [y] B = [z] ) (A = [w] B = [v] ))) → ([y] = [w] [z] = [v] ))
13 th3qlem1.1 . . . . . . . . . . . 12 Er 𝑆
1413a1i 9 . . . . . . . . . . 11 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → Er 𝑆)
15 simprl 471 . . . . . . . . . . . . 13 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → [y] = [w] )
16 erdm 6023 . . . . . . . . . . . . . . . 16 ( Er 𝑆 → dom = 𝑆)
1713, 16ax-mp 7 . . . . . . . . . . . . . . 15 dom = 𝑆
18 simpll 469 . . . . . . . . . . . . . . 15 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → [y] (𝑆 / ))
19 ecelqsdm 6083 . . . . . . . . . . . . . . 15 ((dom = 𝑆 [y] (𝑆 / )) → y 𝑆)
2017, 18, 19sylancr 395 . . . . . . . . . . . . . 14 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → y 𝑆)
2114, 20erth 6057 . . . . . . . . . . . . 13 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → (y w ↔ [y] = [w] ))
2215, 21mpbird 156 . . . . . . . . . . . 12 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → y w)
23 simprr 472 . . . . . . . . . . . . 13 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → [z] = [v] )
24 simplr 470 . . . . . . . . . . . . . . 15 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → [z] (𝑆 / ))
25 ecelqsdm 6083 . . . . . . . . . . . . . . 15 ((dom = 𝑆 [z] (𝑆 / )) → z 𝑆)
2617, 24, 25sylancr 395 . . . . . . . . . . . . . 14 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → z 𝑆)
2714, 26erth 6057 . . . . . . . . . . . . 13 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → (z v ↔ [z] = [v] ))
2823, 27mpbird 156 . . . . . . . . . . . 12 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → z v)
2915, 18eqeltrrd 2093 . . . . . . . . . . . . . 14 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → [w] (𝑆 / ))
30 ecelqsdm 6083 . . . . . . . . . . . . . 14 ((dom = 𝑆 [w] (𝑆 / )) → w 𝑆)
3117, 29, 30sylancr 395 . . . . . . . . . . . . 13 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → w 𝑆)
3223, 24eqeltrrd 2093 . . . . . . . . . . . . . 14 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → [v] (𝑆 / ))
33 ecelqsdm 6083 . . . . . . . . . . . . . 14 ((dom = 𝑆 [v] (𝑆 / )) → v 𝑆)
3417, 32, 33sylancr 395 . . . . . . . . . . . . 13 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → v 𝑆)
35 th3qlem1.3 . . . . . . . . . . . . 13 (((y 𝑆 w 𝑆) (z 𝑆 v 𝑆)) → ((y w z v) → (y + z) (w + v)))
3620, 31, 26, 34, 35syl22anc 1120 . . . . . . . . . . . 12 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → ((y w z v) → (y + z) (w + v)))
3722, 28, 36mp2and 411 . . . . . . . . . . 11 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → (y + z) (w + v))
3814, 37erthi 6059 . . . . . . . . . 10 ((([y] (𝑆 / ) [z] (𝑆 / )) ([y] = [w] [z] = [v] )) → [(y + z)] = [(w + v)] )
397, 12, 38syl2anc 393 . . . . . . . . 9 (((A (𝑆 / ) B (𝑆 / )) ((A = [y] B = [z] ) (A = [w] B = [v] ))) → [(y + z)] = [(w + v)] )
40 eqeq12 2030 . . . . . . . . 9 ((x = [(y + z)] u = [(w + v)] ) → (x = u ↔ [(y + z)] = [(w + v)] ))
4139, 40syl5ibrcom 146 . . . . . . . 8 (((A (𝑆 / ) B (𝑆 / )) ((A = [y] B = [z] ) (A = [w] B = [v] ))) → ((x = [(y + z)] u = [(w + v)] ) → x = u))
4241expimpd 345 . . . . . . 7 ((A (𝑆 / ) B (𝑆 / )) → ((((A = [y] B = [z] ) (A = [w] B = [v] )) (x = [(y + z)] u = [(w + v)] )) → x = u))
432, 42syl5bi 141 . . . . . 6 ((A (𝑆 / ) B (𝑆 / )) → ((((A = [y] B = [z] ) x = [(y + z)] ) ((A = [w] B = [v] ) u = [(w + v)] )) → x = u))
4443exlimdvv 1755 . . . . 5 ((A (𝑆 / ) B (𝑆 / )) → (wv(((A = [y] B = [z] ) x = [(y + z)] ) ((A = [w] B = [v] ) u = [(w + v)] )) → x = u))
4544exlimdvv 1755 . . . 4 ((A (𝑆 / ) B (𝑆 / )) → (yzwv(((A = [y] B = [z] ) x = [(y + z)] ) ((A = [w] B = [v] ) u = [(w + v)] )) → x = u))
461, 45syl5bir 142 . . 3 ((A (𝑆 / ) B (𝑆 / )) → ((yz((A = [y] B = [z] ) x = [(y + z)] ) wv((A = [w] B = [v] ) u = [(w + v)] )) → x = u))
4746alrimivv 1733 . 2 ((A (𝑆 / ) B (𝑆 / )) → xu((yz((A = [y] B = [z] ) x = [(y + z)] ) wv((A = [w] B = [v] ) u = [(w + v)] )) → x = u))
48 eqeq1 2024 . . . . . 6 (x = u → (x = [(y + z)] u = [(y + z)] ))
4948anbi2d 440 . . . . 5 (x = u → (((A = [y] B = [z] ) x = [(y + z)] ) ↔ ((A = [y] B = [z] ) u = [(y + z)] )))
50492exbidv 1726 . . . 4 (x = u → (yz((A = [y] B = [z] ) x = [(y + z)] ) ↔ yz((A = [y] B = [z] ) u = [(y + z)] )))
51 eceq1 6048 . . . . . . . 8 (y = w → [y] = [w] )
5251eqeq2d 2029 . . . . . . 7 (y = w → (A = [y] A = [w] ))
53 eceq1 6048 . . . . . . . 8 (z = v → [z] = [v] )
5453eqeq2d 2029 . . . . . . 7 (z = v → (B = [z] B = [v] ))
5552, 54bi2anan9 526 . . . . . 6 ((y = w z = v) → ((A = [y] B = [z] ) ↔ (A = [w] B = [v] )))
56 oveq12 5441 . . . . . . . 8 ((y = w z = v) → (y + z) = (w + v))
5756eceq1d 6049 . . . . . . 7 ((y = w z = v) → [(y + z)] = [(w + v)] )
5857eqeq2d 2029 . . . . . 6 ((y = w z = v) → (u = [(y + z)] u = [(w + v)] ))
5955, 58anbi12d 445 . . . . 5 ((y = w z = v) → (((A = [y] B = [z] ) u = [(y + z)] ) ↔ ((A = [w] B = [v] ) u = [(w + v)] )))
6059cbvex2v 1777 . . . 4 (yz((A = [y] B = [z] ) u = [(y + z)] ) ↔ wv((A = [w] B = [v] ) u = [(w + v)] ))
6150, 60syl6bb 185 . . 3 (x = u → (yz((A = [y] B = [z] ) x = [(y + z)] ) ↔ wv((A = [w] B = [v] ) u = [(w + v)] )))
6261mo4 1939 . 2 (∃*xyz((A = [y] B = [z] ) x = [(y + z)] ) ↔ xu((yz((A = [y] B = [z] ) x = [(y + z)] ) wv((A = [w] B = [v] ) u = [(w + v)] )) → x = u))
6347, 62sylibr 137 1 ((A (𝑆 / ) B (𝑆 / )) → ∃*xyz((A = [y] B = [z] ) x = [(y + z)] ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224   = wceq 1226  wex 1358   wcel 1370  ∃*wmo 1879   class class class wbr 3734  dom cdm 4268  (class class class)co 5432   Er wer 6010  [cec 6011   / cqs 6012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fv 4833  df-ov 5435  df-er 6013  df-ec 6015  df-qs 6019
This theorem is referenced by:  th3qlem2  6116
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