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Theorem axrepnd 10004
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 10003 . . . 4 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
2 nfnae 2448 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
3 nfnae 2448 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
42, 3nfan 1891 . . . . . 6 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
5 nfnae 2448 . . . . . 6 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
64, 5nfan 1891 . . . . 5 𝑥((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧)
7 nfnae 2448 . . . . . . . . 9 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
8 nfnae 2448 . . . . . . . . 9 𝑧 ¬ ∀𝑥 𝑥 = 𝑧
97, 8nfan 1891 . . . . . . . 8 𝑧(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
10 nfnae 2448 . . . . . . . 8 𝑧 ¬ ∀𝑦 𝑦 = 𝑧
119, 10nfan 1891 . . . . . . 7 𝑧((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧)
12 nfcvf 3004 . . . . . . . . . . . 12 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
1312adantl 482 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑧)
14 nfcvf2 3005 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
1514ad2antrr 722 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑥)
1613, 15nfeld 2986 . . . . . . . . . 10 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑧𝑥)
1716nf5rd 2186 . . . . . . . . 9 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑧𝑥 → ∀𝑦 𝑧𝑥))
18 sp 2172 . . . . . . . . 9 (∀𝑦 𝑧𝑥𝑧𝑥)
1917, 18impbid1 226 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑧𝑥 ↔ ∀𝑦 𝑧𝑥))
20 nfcvf2 3005 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑧𝑧𝑥)
2120ad2antlr 723 . . . . . . . . . . . . 13 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑥)
22 nfcvf2 3005 . . . . . . . . . . . . . 14 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑦)
2322adantl 482 . . . . . . . . . . . . 13 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑦)
2421, 23nfeld 2986 . . . . . . . . . . . 12 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑥𝑦)
2524nf5rd 2186 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑥𝑦 → ∀𝑧 𝑥𝑦))
26 sp 2172 . . . . . . . . . . 11 (∀𝑧 𝑥𝑦𝑥𝑦)
2725, 26impbid1 226 . . . . . . . . . 10 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑥𝑦 ↔ ∀𝑧 𝑥𝑦))
2827anbi1d 629 . . . . . . . . 9 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ((𝑥𝑦 ∧ ∀𝑦𝜑) ↔ (∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
296, 28exbid 2215 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑) ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
3019, 29bibi12d 347 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ((𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ (∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
3111, 30albid 2214 . . . . . 6 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
3231imbi2d 342 . . . . 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 2215 . . . 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 233 . . 3 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
3534exp31 420 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))))
36 nfae 2447 . . . . 5 𝑧𝑥 𝑥 = 𝑦
37 nd2 9998 . . . . . . 7 (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑧𝑥)
3837aecoms 2442 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑧𝑥)
39 nfae 2447 . . . . . . 7 𝑥𝑥 𝑥 = 𝑦
40 nd3 9999 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)
4140intnanrd 490 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ¬ (∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
4239, 41nexd 2213 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ¬ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
4338, 422falsed 378 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
4436, 43alrimi 2203 . . . 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))))
464519.8ad 2171 . 2 (∀𝑥 𝑥 = 𝑦 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
47 nfae 2447 . . . . 5 𝑧𝑥 𝑥 = 𝑧
48 nd4 10000 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑦 𝑧𝑥)
49 nfae 2447 . . . . . . 7 𝑥𝑥 𝑥 = 𝑧
50 nd1 9997 . . . . . . . . 9 (∀𝑧 𝑧 = 𝑥 → ¬ ∀𝑧 𝑥𝑦)
5150aecoms 2442 . . . . . . . 8 (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑧 𝑥𝑦)
5251intnanrd 490 . . . . . . 7 (∀𝑥 𝑥 = 𝑧 → ¬ (∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
5349, 52nexd 2213 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ¬ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
5448, 532falsed 378 . . . . 5 (∀𝑥 𝑥 = 𝑧 → (∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
5547, 54alrimi 2203 . . . 4 (∀𝑥 𝑥 = 𝑧 → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
5655a1d 25 . . 3 (∀𝑥 𝑥 = 𝑧 → (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
575619.8ad 2171 . 2 (∀𝑥 𝑥 = 𝑧 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
58 nfae 2447 . . . . 5 𝑧𝑦 𝑦 = 𝑧
59 nd1 9997 . . . . . 6 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑦 𝑧𝑥)
60 nfae 2447 . . . . . . 7 𝑥𝑦 𝑦 = 𝑧
61 nd2 9998 . . . . . . . . 9 (∀𝑧 𝑧 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)
6261aecoms 2442 . . . . . . . 8 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑧 𝑥𝑦)
6362intnanrd 490 . . . . . . 7 (∀𝑦 𝑦 = 𝑧 → ¬ (∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
6460, 63nexd 2213 . . . . . 6 (∀𝑦 𝑦 = 𝑧 → ¬ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))
6559, 642falsed 378 . . . . 5 (∀𝑦 𝑦 = 𝑧 → (∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
6658, 65alrimi 2203 . . . 4 (∀𝑦 𝑦 = 𝑧 → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
6766a1d 25 . . 3 (∀𝑦 𝑦 = 𝑧 → (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
686719.8ad 2171 . 2 (∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))))
6935, 46, 57, 68pm2.61iii 186 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 207  wa 396  wal 1526  wex 1771  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-reg 9044
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-dif 3936  df-un 3938  df-nul 4289  df-sn 4558  df-pr 4560
This theorem is referenced by:  zfcndrep  10024  axrepprim  32825
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