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Theorem axunndlem1 9629
Description: Lemma for the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
axunndlem1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

Proof of Theorem axunndlem1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 en2lp 8677 . . . . . . . 8 ¬ (𝑦𝑥𝑥𝑦)
2 elequ2 2153 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
32anbi2d 742 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑦𝑥𝑥𝑦) ↔ (𝑦𝑥𝑥𝑧)))
41, 3mtbii 315 . . . . . . 7 (𝑦 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
54sps 2202 . . . . . 6 (∀𝑦 𝑦 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
65nexdv 2013 . . . . 5 (∀𝑦 𝑦 = 𝑧 → ¬ ∃𝑥(𝑦𝑥𝑥𝑧))
76pm2.21d 118 . . . 4 (∀𝑦 𝑦 = 𝑧 → (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
87axc4i 2278 . . 3 (∀𝑦 𝑦 = 𝑧 → ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
9 19.8a 2199 . . 3 (∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥) → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
108, 9syl 17 . 2 (∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
11 zfun 7116 . . 3 𝑥𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥)
12 nfnae 2460 . . . . 5 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
13 nfnae 2460 . . . . . . 7 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
14 nfvd 1993 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤𝑥)
15 nfcvf 2926 . . . . . . . . 9 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
1615nfcrd 2909 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥𝑧)
1714, 16nfand 1975 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑤𝑥𝑥𝑧))
1813, 17nfexd 2312 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑥(𝑤𝑥𝑥𝑧))
1918, 14nfimd 1972 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥))
20 elequ1 2146 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
2120anbi1d 743 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑤𝑥𝑥𝑧) ↔ (𝑦𝑥𝑥𝑧)))
2221exbidv 1999 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑥(𝑤𝑥𝑥𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧)))
2322, 20imbi12d 333 . . . . . 6 (𝑤 = 𝑦 → ((∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
2423a1i 11 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑤 = 𝑦 → ((∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))))
2512, 19, 24cbvald 2422 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
2625exbidv 1999 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑥𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
2711, 26mpbii 223 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
2810, 27pm2.61i 176 1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1630  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115  ax-reg 8664
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-eprel 5179  df-fr 5225
This theorem is referenced by:  axunnd  9630
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