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Theorem axrepndlem1 9452
Description: Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
axrepndlem1 (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
Distinct variable groups:   𝑥,𝑧   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axrepndlem1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 axrep2 4806 . 2 𝑥(∃𝑦𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) → ∀𝑤(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)))
2 nfnae 2351 . . 3 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
3 nfnae 2351 . . . . 5 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
4 nfnae 2351 . . . . . 6 𝑧 ¬ ∀𝑦 𝑦 = 𝑧
5 nfs1v 2465 . . . . . . . 8 𝑧[𝑤 / 𝑧]𝜑
65a1i 11 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧[𝑤 / 𝑧]𝜑)
7 nfcvd 2794 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑤)
8 nfcvf2 2818 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑦)
97, 8nfeqd 2801 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧 𝑤 = 𝑦)
106, 9nfimd 1863 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧([𝑤 / 𝑧]𝜑𝑤 = 𝑦))
11 sbequ12r 2150 . . . . . . . 8 (𝑤 = 𝑧 → ([𝑤 / 𝑧]𝜑𝜑))
12 equequ1 1998 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤 = 𝑦𝑧 = 𝑦))
1311, 12imbi12d 333 . . . . . . 7 (𝑤 = 𝑧 → (([𝑤 / 𝑧]𝜑𝑤 = 𝑦) ↔ (𝜑𝑧 = 𝑦)))
1413a1i 11 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑤 = 𝑧 → (([𝑤 / 𝑧]𝜑𝑤 = 𝑦) ↔ (𝜑𝑧 = 𝑦))))
154, 10, 14cbvald 2313 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) ↔ ∀𝑧(𝜑𝑧 = 𝑦)))
163, 15exbid 2129 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑦𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) ↔ ∃𝑦𝑧(𝜑𝑧 = 𝑦)))
17 nfvd 1884 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧 𝑤𝑥)
188nfcrd 2800 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧 𝑥𝑦)
193, 6nfald 2201 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑦[𝑤 / 𝑧]𝜑)
2018, 19nfand 1866 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑))
212, 20nfexd 2203 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑))
2217, 21nfbid 1872 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)))
23 elequ1 2037 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
2423adantl 481 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → (𝑤𝑥𝑧𝑥))
25 nfeqf2 2333 . . . . . . . . . . 11 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧)
263, 25nfan1 2106 . . . . . . . . . 10 𝑦(¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧)
2711adantl 481 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → ([𝑤 / 𝑧]𝜑𝜑))
2826, 27albid 2128 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → (∀𝑦[𝑤 / 𝑧]𝜑 ↔ ∀𝑦𝜑))
2928anbi2d 740 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → ((𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑) ↔ (𝑥𝑦 ∧ ∀𝑦𝜑)))
3029exbidv 1890 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → (∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
3124, 30bibi12d 334 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → ((𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
3231ex 449 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑤 = 𝑧 → ((𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
334, 22, 32cbvald 2313 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
3416, 33imbi12d 333 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → ((∃𝑦𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) → ∀𝑤(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑))) ↔ (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
352, 34exbid 2129 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑥(∃𝑦𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) → ∀𝑤(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑))) ↔ ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
361, 35mpbii 223 1 (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521  wex 1744  wnf 1748  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-cleq 2644  df-clel 2647  df-nfc 2782
This theorem is referenced by:  axrepndlem2  9453
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