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Theorem axrepndlem1 10008
Description: Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2386. (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 5186 . 2 𝑥(∃𝑦𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) → ∀𝑤(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)))
2 nfnae 2452 . . 3 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
3 nfnae 2452 . . . . 5 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
4 nfnae 2452 . . . . . 6 𝑧 ¬ ∀𝑦 𝑦 = 𝑧
5 nfs1v 2269 . . . . . . . 8 𝑧[𝑤 / 𝑧]𝜑
65a1i 11 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧[𝑤 / 𝑧]𝜑)
7 nfcvd 2978 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑤)
8 nfcvf2 3008 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑦)
97, 8nfeqd 2988 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧 𝑤 = 𝑦)
106, 9nfimd 1891 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧([𝑤 / 𝑧]𝜑𝑤 = 𝑦))
11 sbequ12r 2249 . . . . . . . 8 (𝑤 = 𝑧 → ([𝑤 / 𝑧]𝜑𝜑))
12 equequ1 2028 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤 = 𝑦𝑧 = 𝑦))
1311, 12imbi12d 347 . . . . . . 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 2424 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) ↔ ∀𝑧(𝜑𝑧 = 𝑦)))
163, 15exbid 2220 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑦𝑤([𝑤 / 𝑧]𝜑𝑤 = 𝑦) ↔ ∃𝑦𝑧(𝜑𝑧 = 𝑦)))
17 nfvd 1912 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧 𝑤𝑥)
188nfcrd 2969 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧 𝑥𝑦)
193, 6nfald 2343 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑦[𝑤 / 𝑧]𝜑)
2018, 19nfand 1894 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑))
212, 20nfexd 2344 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑))
2217, 21nfbid 1899 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)))
23 elequ1 2117 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
2423adantl 484 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → (𝑤𝑥𝑧𝑥))
25 nfeqf2 2391 . . . . . . . . . . 11 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧)
263, 25nfan1 2195 . . . . . . . . . 10 𝑦(¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧)
2711adantl 484 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → ([𝑤 / 𝑧]𝜑𝜑))
2826, 27albid 2219 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → (∀𝑦[𝑤 / 𝑧]𝜑 ↔ ∀𝑦𝜑))
2928anbi2d 630 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → ((𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑) ↔ (𝑥𝑦 ∧ ∀𝑦𝜑)))
3029exbidv 1918 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → (∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
3124, 30bibi12d 348 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑧𝑤 = 𝑧) → ((𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
3231ex 415 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑤 = 𝑧 → ((𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
334, 22, 32cbvald 2424 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤(𝑤𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦[𝑤 / 𝑧]𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
3416, 33imbi12d 347 . . 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 2220 . 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 235 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 208  wa 398  wal 1531  wex 1776  wnf 1780  [wsb 2065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-13 2386  ax-ext 2793  ax-rep 5183
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-cleq 2814  df-nfc 2963
This theorem is referenced by:  axrepndlem2  10009
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