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Theorem axsepg 5224
Description: A more general version of the axiom scheme of separation ax-sep 5223, where variable 𝑧 can also occur (in addition to 𝑥) in formula 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version is derived from the more restrictive ax-sep 5223 with no additional set theory axioms. Note that it was also derived from ax-rep 5209 but without ax-sep 5223 as axsepgfromrep 5221. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2171 and ax-13 2372 and shorten proof. (Revised by BJ, 6-Oct-2019.)
Assertion
Ref Expression
axsepg 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑧)

Proof of Theorem axsepg
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2121 . . . . . 6 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
21anbi1d 630 . . . . 5 (𝑤 = 𝑧 → ((𝑥𝑤𝜑) ↔ (𝑥𝑧𝜑)))
32bibi2d 343 . . . 4 (𝑤 = 𝑧 → ((𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
43albidv 1923 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
54exbidv 1924 . 2 (𝑤 = 𝑧 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
6 ax-sep 5223 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤𝜑))
75, 6chvarvv 2002 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wal 1537  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-sep 5223
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783
This theorem is referenced by: (None)
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