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Theorem axsep2 4047
 Description: A less restrictive version of the Separation Scheme ax-sep 4046, where variables 𝑥 and 𝑧 can both appear free in the wff 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version was derived from the more restrictive ax-sep 4046 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
axsep2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑧)

Proof of Theorem axsep2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2203 . . . . . . 7 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
21anbi1d 460 . . . . . 6 (𝑤 = 𝑧 → ((𝑥𝑤 ∧ (𝑥𝑧𝜑)) ↔ (𝑥𝑧 ∧ (𝑥𝑧𝜑))))
3 anabs5 562 . . . . . 6 ((𝑥𝑧 ∧ (𝑥𝑧𝜑)) ↔ (𝑥𝑧𝜑))
42, 3syl6bb 195 . . . . 5 (𝑤 = 𝑧 → ((𝑥𝑤 ∧ (𝑥𝑧𝜑)) ↔ (𝑥𝑧𝜑)))
54bibi2d 231 . . . 4 (𝑤 = 𝑧 → ((𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑))) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
65albidv 1796 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑))) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
76exbidv 1797 . 2 (𝑤 = 𝑧 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑))) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
8 ax-sep 4046 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑)))
97, 8chvarv 1909 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∀wal 1329  ∃wex 1468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121  ax-sep 4046 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2132  df-clel 2135 This theorem is referenced by: (None)
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