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Theorem th3qlem1 6238
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 1825 . . . 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 528 . . . . . . 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 2116 . . . . . . . . . . . . 13 (𝐴 = [𝑦] → (𝐴 ∈ (𝑆 / ) ↔ [𝑦] ∈ (𝑆 / )))
4 eleq1 2116 . . . . . . . . . . . . 13 (𝐵 = [𝑧] → (𝐵 ∈ (𝑆 / ) ↔ [𝑧] ∈ (𝑆 / )))
53, 4bi2anan9 548 . . . . . . . . . . . 12 ((𝐴 = [𝑦] 𝐵 = [𝑧] ) → ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) ↔ ([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / ))))
65adantr 265 . . . . . . . . . . 11 (((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] )) → ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) ↔ ([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / ))))
76biimpac 286 . . . . . . . . . 10 (((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) ∧ ((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] ))) → ([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )))
8 eqtr2 2074 . . . . . . . . . . . . 13 ((𝐴 = [𝑦] 𝐴 = [𝑤] ) → [𝑦] = [𝑤] )
9 eqtr2 2074 . . . . . . . . . . . . 13 ((𝐵 = [𝑧] 𝐵 = [𝑣] ) → [𝑧] = [𝑣] )
108, 9anim12i 325 . . . . . . . . . . . 12 (((𝐴 = [𝑦] 𝐴 = [𝑤] ) ∧ (𝐵 = [𝑧] 𝐵 = [𝑣] )) → ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] ))
1110an4s 530 . . . . . . . . . . 11 (((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] )) → ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] ))
1211adantl 266 . . . . . . . . . 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 491 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑦] = [𝑤] )
16 erdm 6146 . . . . . . . . . . . . . . . 16 ( Er 𝑆 → dom = 𝑆)
1713, 16ax-mp 7 . . . . . . . . . . . . . . 15 dom = 𝑆
18 simpll 489 . . . . . . . . . . . . . . 15 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑦] ∈ (𝑆 / ))
19 ecelqsdm 6206 . . . . . . . . . . . . . . 15 ((dom = 𝑆 ∧ [𝑦] ∈ (𝑆 / )) → 𝑦𝑆)
2017, 18, 19sylancr 399 . . . . . . . . . . . . . 14 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑦𝑆)
2114, 20erth 6180 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → (𝑦 𝑤 ↔ [𝑦] = [𝑤] ))
2215, 21mpbird 160 . . . . . . . . . . . 12 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑦 𝑤)
23 simprr 492 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑧] = [𝑣] )
24 simplr 490 . . . . . . . . . . . . . . 15 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑧] ∈ (𝑆 / ))
25 ecelqsdm 6206 . . . . . . . . . . . . . . 15 ((dom = 𝑆 ∧ [𝑧] ∈ (𝑆 / )) → 𝑧𝑆)
2617, 24, 25sylancr 399 . . . . . . . . . . . . . 14 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑧𝑆)
2714, 26erth 6180 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → (𝑧 𝑣 ↔ [𝑧] = [𝑣] ))
2823, 27mpbird 160 . . . . . . . . . . . 12 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑧 𝑣)
2915, 18eqeltrrd 2131 . . . . . . . . . . . . . 14 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑤] ∈ (𝑆 / ))
30 ecelqsdm 6206 . . . . . . . . . . . . . 14 ((dom = 𝑆 ∧ [𝑤] ∈ (𝑆 / )) → 𝑤𝑆)
3117, 29, 30sylancr 399 . . . . . . . . . . . . 13 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → 𝑤𝑆)
3223, 24eqeltrrd 2131 . . . . . . . . . . . . . 14 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [𝑣] ∈ (𝑆 / ))
33 ecelqsdm 6206 . . . . . . . . . . . . . 14 ((dom = 𝑆 ∧ [𝑣] ∈ (𝑆 / )) → 𝑣𝑆)
3417, 32, 33sylancr 399 . . . . . . . . . . . . 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 1147 . . . . . . . . . . . 12 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → ((𝑦 𝑤𝑧 𝑣) → (𝑦 + 𝑧) (𝑤 + 𝑣)))
3722, 28, 36mp2and 417 . . . . . . . . . . 11 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → (𝑦 + 𝑧) (𝑤 + 𝑣))
3814, 37erthi 6182 . . . . . . . . . 10 ((([𝑦] ∈ (𝑆 / ) ∧ [𝑧] ∈ (𝑆 / )) ∧ ([𝑦] = [𝑤] ∧ [𝑧] = [𝑣] )) → [(𝑦 + 𝑧)] = [(𝑤 + 𝑣)] )
397, 12, 38syl2anc 397 . . . . . . . . 9 (((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) ∧ ((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] ))) → [(𝑦 + 𝑧)] = [(𝑤 + 𝑣)] )
40 eqeq12 2068 . . . . . . . . 9 ((𝑥 = [(𝑦 + 𝑧)] 𝑢 = [(𝑤 + 𝑣)] ) → (𝑥 = 𝑢 ↔ [(𝑦 + 𝑧)] = [(𝑤 + 𝑣)] ))
4139, 40syl5ibrcom 150 . . . . . . . 8 (((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) ∧ ((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] ))) → ((𝑥 = [(𝑦 + 𝑧)] 𝑢 = [(𝑤 + 𝑣)] ) → 𝑥 = 𝑢))
4241expimpd 349 . . . . . . 7 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → ((((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ (𝐴 = [𝑤] 𝐵 = [𝑣] )) ∧ (𝑥 = [(𝑦 + 𝑧)] 𝑢 = [(𝑤 + 𝑣)] )) → 𝑥 = 𝑢))
432, 42syl5bi 145 . . . . . 6 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → ((((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ∧ ((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )) → 𝑥 = 𝑢))
4443exlimdvv 1793 . . . . 5 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → (∃𝑤𝑣(((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ∧ ((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )) → 𝑥 = 𝑢))
4544exlimdvv 1793 . . . 4 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → (∃𝑦𝑧𝑤𝑣(((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ∧ ((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )) → 𝑥 = 𝑢))
461, 45syl5bir 146 . . 3 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → ((∃𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ∧ ∃𝑤𝑣((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )) → 𝑥 = 𝑢))
4746alrimivv 1771 . 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 2062 . . . . . 6 (𝑥 = 𝑢 → (𝑥 = [(𝑦 + 𝑧)] 𝑢 = [(𝑦 + 𝑧)] ))
4948anbi2d 445 . . . . 5 (𝑥 = 𝑢 → (((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ↔ ((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑢 = [(𝑦 + 𝑧)] )))
50492exbidv 1764 . . . 4 (𝑥 = 𝑢 → (∃𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ↔ ∃𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑢 = [(𝑦 + 𝑧)] )))
51 eceq1 6171 . . . . . . . 8 (𝑦 = 𝑤 → [𝑦] = [𝑤] )
5251eqeq2d 2067 . . . . . . 7 (𝑦 = 𝑤 → (𝐴 = [𝑦] 𝐴 = [𝑤] ))
53 eceq1 6171 . . . . . . . 8 (𝑧 = 𝑣 → [𝑧] = [𝑣] )
5453eqeq2d 2067 . . . . . . 7 (𝑧 = 𝑣 → (𝐵 = [𝑧] 𝐵 = [𝑣] ))
5552, 54bi2anan9 548 . . . . . 6 ((𝑦 = 𝑤𝑧 = 𝑣) → ((𝐴 = [𝑦] 𝐵 = [𝑧] ) ↔ (𝐴 = [𝑤] 𝐵 = [𝑣] )))
56 oveq12 5548 . . . . . . . 8 ((𝑦 = 𝑤𝑧 = 𝑣) → (𝑦 + 𝑧) = (𝑤 + 𝑣))
5756eceq1d 6172 . . . . . . 7 ((𝑦 = 𝑤𝑧 = 𝑣) → [(𝑦 + 𝑧)] = [(𝑤 + 𝑣)] )
5857eqeq2d 2067 . . . . . 6 ((𝑦 = 𝑤𝑧 = 𝑣) → (𝑢 = [(𝑦 + 𝑧)] 𝑢 = [(𝑤 + 𝑣)] ))
5955, 58anbi12d 450 . . . . 5 ((𝑦 = 𝑤𝑧 = 𝑣) → (((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑢 = [(𝑦 + 𝑧)] ) ↔ ((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )))
6059cbvex2v 1815 . . . 4 (∃𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑢 = [(𝑦 + 𝑧)] ) ↔ ∃𝑤𝑣((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] ))
6150, 60syl6bb 189 . . 3 (𝑥 = 𝑢 → (∃𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ) ↔ ∃𝑤𝑣((𝐴 = [𝑤] 𝐵 = [𝑣] ) ∧ 𝑢 = [(𝑤 + 𝑣)] )))
6261mo4 1977 . 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 141 1 ((𝐴 ∈ (𝑆 / ) ∧ 𝐵 ∈ (𝑆 / )) → ∃*𝑥𝑦𝑧((𝐴 = [𝑦] 𝐵 = [𝑧] ) ∧ 𝑥 = [(𝑦 + 𝑧)] ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  ∃*wmo 1917   class class class wbr 3791  dom cdm 4372  (class class class)co 5539   Er wer 6133  [cec 6134   / cqs 6135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fv 4937  df-ov 5542  df-er 6136  df-ec 6138  df-qs 6142
This theorem is referenced by:  th3qlem2  6239
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