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Theorem axrepndlem1 10617
Description: Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
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 5289 . 2 𝑥(∃𝑦𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) → ∀𝑤(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)))
2 nfnae 2427 . . 3 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
3 nfnae 2427 . . . . 5 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
4 nfnae 2427 . . . . . 6 𝑧 ¬ ∀𝑦 𝑦 = 𝑧
5 nfs1v 2145 . . . . . . . 8 𝑧[𝑤 / 𝑧]𝜑
65a1i 11 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧[𝑤 / 𝑧]𝜑)
7 nfcvd 2892 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑤)
8 nfcvf2 2922 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑦)
97, 8nfeqd 2902 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧 𝑤 = 𝑦)
106, 9nfimd 1889 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧([𝑤 / 𝑧]𝜑𝑤 = 𝑦))
11 sbequ12r 2239 . . . . . . . 8 (𝑤 = 𝑧 → ([𝑤 / 𝑧]𝜑𝜑))
12 equequ1 2020 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤 = 𝑦𝑧 = 𝑦))
1311, 12imbi12d 343 . . . . . . 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 2400 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) ↔ ∀𝑧(𝜑𝑧 = 𝑦)))
163, 15exbid 2211 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑦𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) ↔ ∃𝑦𝑧(𝜑𝑧 = 𝑦)))
17 nfvd 1910 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧 𝑤𝑥)
188nfcrd 2884 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧 𝑥𝑦)
193, 6nfald 2316 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑦[𝑤 / 𝑧]𝜑)
2018, 19nfand 1892 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑))
212, 20nfexd 2317 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑))
2217, 21nfbid 1897 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)))
23 elequ1 2105 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
2423adantl 480 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → (𝑤𝑥𝑧𝑥))
25 nfeqf2 2370 . . . . . . . . . . 11 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧)
263, 25nfan1 2188 . . . . . . . . . 10 𝑦(¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧)
2711adantl 480 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → ([𝑤 / 𝑧]𝜑𝜑))
2826, 27albid 2210 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → (∀𝑦[𝑤 / 𝑧]𝜑 ↔ ∀𝑦𝜑))
2928anbi2d 628 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → ((𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑) ↔ (𝑥𝑦 ∧ ∀𝑦𝜑)))
3029exbidv 1916 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → (∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
3124, 30bibi12d 344 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → ((𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
3231ex 411 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑤 = 𝑧 → ((𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
334, 22, 32cbvald 2400 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
3416, 33imbi12d 343 . . 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 2211 . 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 232 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 205  wa 394  wal 1531  wex 1773  wnf 1777  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2365  ax-ext 2696  ax-rep 5286
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-cleq 2717  df-clel 2802  df-nfc 2877
This theorem is referenced by:  axrepndlem2  10618
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