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Theorem th3qlem1 6663
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 (((𝑦𝑆𝑤𝑆) ∧ (𝑧𝑆𝑣𝑆)) → ((𝑦 𝑤𝑧 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 1946 . . . 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 586 . . . . . . 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 2252 . . . . . . . . . . . . 13 (𝐴 = [𝑦] → (𝐴 ∈ (𝑆 / ) ↔ [𝑦] ∈ (𝑆 / )))
4 eleq1 2252 . . . . . . . . . . . . 13 (𝐵 = [𝑧] → (𝐵 ∈ (𝑆 / ) ↔ [𝑧] ∈ (𝑆 / )))
53, 4bi2anan9 606 . . . . . . . . . . . 12 ((𝐴 = [𝑦] 𝐵 = [𝑧] ) → ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) ↔ ([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / ))))
65adantr 276 . . . . . . . . . . 11 (((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] )) → ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) ↔ ([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / ))))
76biimpac 298 . . . . . . . . . 10 (((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) ∧ ((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] ))) → ([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )))
8 eqtr2 2208 . . . . . . . . . . . . 13 ((𝐴 = [𝑦] 𝐴 = [𝑤] ) → [𝑦] = [𝑤] )
9 eqtr2 2208 . . . . . . . . . . . . 13 ((𝐵 = [𝑧] 𝐵 = [𝑣] ) → [𝑧] = [𝑣] )
108, 9anim12i 338 . . . . . . . . . . . 12 (((𝐴 = [𝑦] 𝐴 = [𝑤] ) ∧ (𝐵 = [𝑧] 𝐵 = [𝑣] )) → ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] ))
1110an4s 588 . . . . . . . . . . 11 (((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] )) → ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] ))
1211adantl 277 . . . . . . . . . 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 529 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑦] = [𝑤] )
16 erdm 6569 . . . . . . . . . . . . . . . 16 ( Er 𝑆 → dom = 𝑆)
1713, 16ax-mp 5 . . . . . . . . . . . . . . 15 dom = 𝑆
18 simpll 527 . . . . . . . . . . . . . . 15 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑦] ∈ (𝑆 / ))
19 ecelqsdm 6631 . . . . . . . . . . . . . . 15 ((dom = 𝑆 ∧ [𝑦] ∈ (𝑆 / )) → 𝑦𝑆)
2017, 18, 19sylancr 414 . . . . . . . . . . . . . 14 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑦𝑆)
2114, 20erth 6605 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → (𝑦 𝑤 ↔ [𝑦] = [𝑤] ))
2215, 21mpbird 167 . . . . . . . . . . . 12 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑦 𝑤)
23 simprr 531 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑧] = [𝑣] )
24 simplr 528 . . . . . . . . . . . . . . 15 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑧] ∈ (𝑆 / ))
25 ecelqsdm 6631 . . . . . . . . . . . . . . 15 ((dom = 𝑆 ∧ [𝑧] ∈ (𝑆 / )) → 𝑧𝑆)
2617, 24, 25sylancr 414 . . . . . . . . . . . . . 14 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑧𝑆)
2714, 26erth 6605 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → (𝑧 𝑣 ↔ [𝑧] = [𝑣] ))
2823, 27mpbird 167 . . . . . . . . . . . 12 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑧 𝑣)
2915, 18eqeltrrd 2267 . . . . . . . . . . . . . 14 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑤] ∈ (𝑆 / ))
30 ecelqsdm 6631 . . . . . . . . . . . . . 14 ((dom = 𝑆 ∧ [𝑤] ∈ (𝑆 / )) → 𝑤𝑆)
3117, 29, 30sylancr 414 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑤𝑆)
3223, 24eqeltrrd 2267 . . . . . . . . . . . . . 14 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑣] ∈ (𝑆 / ))
33 ecelqsdm 6631 . . . . . . . . . . . . . 14 ((dom = 𝑆 ∧ [𝑣] ∈ (𝑆 / )) → 𝑣𝑆)
3417, 32, 33sylancr 414 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑣𝑆)
35 th3qlem1.3 . . . . . . . . . . . . 13 (((𝑦𝑆𝑤𝑆) ∧ (𝑧𝑆𝑣𝑆)) → ((𝑦 𝑤𝑧 𝑣) → (𝑦 + 𝑧) (𝑤 + 𝑣)))
3620, 31, 26, 34, 35syl22anc 1250 . . . . . . . . . . . 12 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → ((𝑦 𝑤𝑧 𝑣) → (𝑦 + 𝑧) (𝑤 + 𝑣)))
3722, 28, 36mp2and 433 . . . . . . . . . . 11 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → (𝑦 + 𝑧) (𝑤 + 𝑣))
3814, 37erthi 6607 . . . . . . . . . 10 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [(𝑦 + 𝑧)] = [(𝑤 + 𝑣)] )
397, 12, 38syl2anc 411 . . . . . . . . 9 (((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) ∧ ((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] ))) → [(𝑦 + 𝑧)] = [(𝑤 + 𝑣)] )
40 eqeq12 2202 . . . . . . . . 9 ((𝑥 = [(𝑦 + 𝑧)] 𝑢 = [(𝑤 + 𝑣)] ) → (𝑥 = 𝑢 ↔ [(𝑦 + 𝑧)] = [(𝑤 + 𝑣)] ))
4139, 40syl5ibrcom 157 . . . . . . . 8 (((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) ∧ ((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] ))) → ((𝑥 = [(𝑦 + 𝑧)] 𝑢 = [(𝑤 + 𝑣)] ) → 𝑥 = 𝑢))
4241expimpd 363 . . . . . . 7 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → ((((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] )) ∧ (𝑥 = [(𝑦 + 𝑧)] 𝑢 = [(𝑤 + 𝑣)] )) → 𝑥 = 𝑢))
432, 42biimtrid 152 . . . . . 6 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → ((((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ∧ ((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )) → 𝑥 = 𝑢))
4443exlimdvv 1909 . . . . 5 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → (∃𝑤𝑣(((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ∧ ((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )) → 𝑥 = 𝑢))
4544exlimdvv 1909 . . . 4 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → (∃𝑦𝑧𝑤𝑣(((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ∧ ((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )) → 𝑥 = 𝑢))
461, 45biimtrrid 153 . . 3 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → ((∃𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ∧ ∃𝑤𝑣((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )) → 𝑥 = 𝑢))
4746alrimivv 1886 . 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 2196 . . . . . 6 (𝑥 = 𝑢 → (𝑥 = [(𝑦 + 𝑧)] 𝑢 = [(𝑦 + 𝑧)] ))
4948anbi2d 464 . . . . 5 (𝑥 = 𝑢 → (((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ↔ ((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑢 = [(𝑦 + 𝑧)] )))
50492exbidv 1879 . . . 4 (𝑥 = 𝑢 → (∃𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ↔ ∃𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑢 = [(𝑦 + 𝑧)] )))
51 eceq1 6594 . . . . . . . 8 (𝑦 = 𝑤 → [𝑦] = [𝑤] )
5251eqeq2d 2201 . . . . . . 7 (𝑦 = 𝑤 → (𝐴 = [𝑦] 𝐴 = [𝑤] ))
53 eceq1 6594 . . . . . . . 8 (𝑧 = 𝑣 → [𝑧] = [𝑣] )
5453eqeq2d 2201 . . . . . . 7 (𝑧 = 𝑣 → (𝐵 = [𝑧] 𝐵 = [𝑣] ))
5552, 54bi2anan9 606 . . . . . 6 ((𝑦 = 𝑤𝑧 = 𝑣) → ((𝐴 = [𝑦] 𝐵 = [𝑧] ) ↔ (𝐴 = [𝑤] 𝐵 = [𝑣] )))
56 oveq12 5905 . . . . . . . 8 ((𝑦 = 𝑤𝑧 = 𝑣) → (𝑦 + 𝑧) = (𝑤 + 𝑣))
5756eceq1d 6595 . . . . . . 7 ((𝑦 = 𝑤𝑧 = 𝑣) → [(𝑦 + 𝑧)] = [(𝑤 + 𝑣)] )
5857eqeq2d 2201 . . . . . 6 ((𝑦 = 𝑤𝑧 = 𝑣) → (𝑢 = [(𝑦 + 𝑧)] 𝑢 = [(𝑤 + 𝑣)] ))
5955, 58anbi12d 473 . . . . 5 ((𝑦 = 𝑤𝑧 = 𝑣) → (((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑢 = [(𝑦 + 𝑧)] ) ↔ ((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )))
6059cbvex2v 1936 . . . 4 (∃𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑢 = [(𝑦 + 𝑧)] ) ↔ ∃𝑤𝑣((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] ))
6150, 60bitrdi 196 . . 3 (𝑥 = 𝑢 → (∃𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ↔ ∃𝑤𝑣((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )))
6261mo4 2099 . 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 134 1 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → ∃*𝑥𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362   = wceq 1364  wex 1503  ∃*wmo 2039  wcel 2160   class class class wbr 4018  dom cdm 4644  (class class class)co 5896   Er wer 6556  [cec 6557   / cqs 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fv 5243  df-ov 5899  df-er 6559  df-ec 6561  df-qs 6565
This theorem is referenced by:  th3qlem2  6664
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