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Theorem axsep2 4134
Description: A less restrictive version of the Separation Scheme ax-sep 4133, 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 4133 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 2251 . . . . . . 7 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
21anbi1d 465 . . . . . 6 (𝑤 = 𝑧 → ((𝑥𝑤 ∧ (𝑥𝑧𝜑)) ↔ (𝑥𝑧 ∧ (𝑥𝑧𝜑))))
3 anabs5 573 . . . . . 6 ((𝑥𝑧 ∧ (𝑥𝑧𝜑)) ↔ (𝑥𝑧𝜑))
42, 3bitrdi 196 . . . . 5 (𝑤 = 𝑧 → ((𝑥𝑤 ∧ (𝑥𝑧𝜑)) ↔ (𝑥𝑧𝜑)))
54bibi2d 232 . . . 4 (𝑤 = 𝑧 → ((𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑))) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
65albidv 1834 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑))) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
76exbidv 1835 . 2 (𝑤 = 𝑧 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑))) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
8 ax-sep 4133 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑)))
97, 8chvarv 1947 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wal 1361  wex 1502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-ext 2169  ax-sep 4133
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-cleq 2180  df-clel 2183
This theorem is referenced by: (None)
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