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Theorem axrepnd 9704
Description: A version of the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
axrepnd 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))

Proof of Theorem axrepnd
StepHypRef Expression
1 axrepndlem2 9703 . . . 4 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
2 nfnae 2441 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
3 nfnae 2441 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
42, 3nfan 1999 . . . . . 6 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
5 nfnae 2441 . . . . . 6 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
64, 5nfan 1999 . . . . 5 𝑥((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧)
7 nfnae 2441 . . . . . . . . 9 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
8 nfnae 2441 . . . . . . . . 9 𝑧 ¬ ∀𝑥 𝑥 = 𝑧
97, 8nfan 1999 . . . . . . . 8 𝑧(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
10 nfnae 2441 . . . . . . . 8 𝑧 ¬ ∀𝑦 𝑦 = 𝑧
119, 10nfan 1999 . . . . . . 7 𝑧((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧)
12 nfcvf 2965 . . . . . . . . . . . 12 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
1312adantl 474 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑧)
14 nfcvf2 2966 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
1514ad2antrr 718 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑥)
1613, 15nfeld 2950 . . . . . . . . . 10 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑧𝑥)
1716nf5rd 2230 . . . . . . . . 9 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑧𝑥 → ∀𝑦 𝑧𝑥))
18 sp 2217 . . . . . . . . 9 (∀𝑦 𝑧𝑥𝑧𝑥)
1917, 18impbid1 217 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑧𝑥 ↔ ∀𝑦 𝑧𝑥))
20 nfcvf2 2966 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑧𝑧𝑥)
2120ad2antlr 719 . . . . . . . . . . . . 13 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑥)
22 nfcvf2 2966 . . . . . . . . . . . . . 14 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑦)
2322adantl 474 . . . . . . . . . . . . 13 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑦)
2421, 23nfeld 2950 . . . . . . . . . . . 12 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑥𝑦)
2524nf5rd 2230 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑥𝑦 → ∀𝑧 𝑥𝑦))
26 sp 2217 . . . . . . . . . . 11 (∀𝑧 𝑥𝑦𝑥𝑦)
2725, 26impbid1 217 . . . . . . . . . 10 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑥𝑦 ↔ ∀𝑧 𝑥𝑦))
2827anbi1d 624 . . . . . . . . 9 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ((𝑥𝑦 ∧ ∀𝑦𝜑) ↔ (∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
296, 28exbid 2258 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑) ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
3019, 29bibi12d 337 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ((𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ (∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
3111, 30albid 2257 . . . . . 6 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
3231imbi2d 332 . . . . 5 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ((∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))))
336, 32exbid 2258 . . . 4 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))))
341, 33mpbid 224 . . 3 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
3534exp31 411 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))))
36 nfae 2439 . . . . 5 𝑧𝑥 𝑥 = 𝑦
37 nd2 9698 . . . . . . 7 (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑧𝑥)
3837aecoms 2435 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑧𝑥)
39 nfae 2439 . . . . . . 7 𝑥𝑥 𝑥 = 𝑦
40 nd3 9699 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)
4140intnanrd 484 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ¬ (∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
4239, 41nexd 2256 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ¬ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
4338, 422falsed 368 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
4436, 43alrimi 2248 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
4544a1d 25 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
46 19.8a 2216 . . 3 ((∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))) → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
4745, 46syl 17 . 2 (∀𝑥 𝑥 = 𝑦 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
48 nfae 2439 . . . . 5 𝑧𝑥 𝑥 = 𝑧
49 nd4 9700 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑦 𝑧𝑥)
50 nfae 2439 . . . . . . 7 𝑥𝑥 𝑥 = 𝑧
51 nd1 9697 . . . . . . . . 9 (∀𝑧 𝑧 = 𝑥 → ¬ ∀𝑧 𝑥𝑦)
5251aecoms 2435 . . . . . . . 8 (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑧 𝑥𝑦)
5352intnanrd 484 . . . . . . 7 (∀𝑥 𝑥 = 𝑧 → ¬ (∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
5450, 53nexd 2256 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ¬ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
5549, 542falsed 368 . . . . 5 (∀𝑥 𝑥 = 𝑧 → (∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
5648, 55alrimi 2248 . . . 4 (∀𝑥 𝑥 = 𝑧 → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
5756a1d 25 . . 3 (∀𝑥 𝑥 = 𝑧 → (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
5857, 46syl 17 . 2 (∀𝑥 𝑥 = 𝑧 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
59 nfae 2439 . . . . 5 𝑧𝑦 𝑦 = 𝑧
60 nd1 9697 . . . . . 6 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑦 𝑧𝑥)
61 nfae 2439 . . . . . . 7 𝑥𝑦 𝑦 = 𝑧
62 nd2 9698 . . . . . . . . 9 (∀𝑧 𝑧 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)
6362aecoms 2435 . . . . . . . 8 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑧 𝑥𝑦)
6463intnanrd 484 . . . . . . 7 (∀𝑦 𝑦 = 𝑧 → ¬ (∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
6561, 64nexd 2256 . . . . . 6 (∀𝑦 𝑦 = 𝑧 → ¬ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
6660, 652falsed 368 . . . . 5 (∀𝑦 𝑦 = 𝑧 → (∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
6759, 66alrimi 2248 . . . 4 (∀𝑦 𝑦 = 𝑧 → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
6867a1d 25 . . 3 (∀𝑦 𝑦 = 𝑧 → (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
6968, 46syl 17 . 2 (∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
7035, 47, 58, 69pm2.61iii 180 1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wal 1651  wex 1875  wnfc 2928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-reg 8739
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-un 3774  df-nul 4116  df-sn 4369  df-pr 4371
This theorem is referenced by:  zfcndrep  9724  axrepprim  32094
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