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Theorem axsepg2 35075
Description: A generalization of ax-sep 5302 in which 𝑦 and 𝑧 need not be distinct. See also axsepg 5303 which instead allows 𝑧 to occur in 𝜑. Usage of this theorem is discouraged because it depends on ax-13 2375. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.)
Assertion
Ref Expression
axsepg2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑧   𝜑,𝑦   𝜑,𝑧   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem axsepg2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1912 . . 3 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
2 nfvd 1913 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥𝑤)
3 nfcvf 2930 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
43nfcrd 2897 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥𝑧)
5 nfvd 1913 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝜑)
64, 5nfand 1895 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑥𝑧𝜑))
72, 6nfbid 1900 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑥𝑤 ↔ (𝑥𝑧𝜑)))
81, 7nfald 2327 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑥(𝑥𝑤 ↔ (𝑥𝑧𝜑)))
9 nfvd 1913 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑤𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)))
10 elequ2 2121 . . . . . 6 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
1110bibi1d 343 . . . . 5 (𝑤 = 𝑦 → ((𝑥𝑤 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
1211biimpd 229 . . . 4 (𝑤 = 𝑦 → ((𝑥𝑤 ↔ (𝑥𝑧𝜑)) → (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
1312alimdv 1914 . . 3 (𝑤 = 𝑦 → (∀𝑥(𝑥𝑤 ↔ (𝑥𝑧𝜑)) → ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
1413a1i 11 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑤 = 𝑦 → (∀𝑥(𝑥𝑤 ↔ (𝑥𝑧𝜑)) → ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)))))
15 elequ2 2121 . . . . . . 7 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
1615anbi1d 631 . . . . . 6 (𝑦 = 𝑧 → ((𝑥𝑦𝜑) ↔ (𝑥𝑧𝜑)))
1716bibi2d 342 . . . . 5 (𝑦 = 𝑧 → ((𝑥𝑦 ↔ (𝑥𝑦𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
1817biimpd 229 . . . 4 (𝑦 = 𝑧 → ((𝑥𝑦 ↔ (𝑥𝑦𝜑)) → (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
1918alimdv 1914 . . 3 (𝑦 = 𝑧 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑦𝜑)) → ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
2019sps 2183 . 2 (∀𝑦 𝑦 = 𝑧 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑦𝜑)) → ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
21 ax-sep 5302 . 2 𝑤𝑥(𝑥𝑤 ↔ (𝑥𝑧𝜑))
22 ax-nul 5312 . . 3 𝑦𝑥 ¬ 𝑥𝑦
23 id 22 . . . . 5 𝑥𝑦 → ¬ 𝑥𝑦)
2423bianfd 534 . . . 4 𝑥𝑦 → (𝑥𝑦 ↔ (𝑥𝑦𝜑)))
2524alimi 1808 . . 3 (∀𝑥 ¬ 𝑥𝑦 → ∀𝑥(𝑥𝑦 ↔ (𝑥𝑦𝜑)))
2622, 25eximii 1834 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑦𝜑))
278, 9, 14, 20, 21, 26dvelimexcasei 35071 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1535  wex 1776
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-sep 5302  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-clel 2814  df-nfc 2890
This theorem is referenced by: (None)
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