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Theorem axsep2 4710
 Description: A less restrictive version of the Separation Scheme axsep 4708, 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 4709 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 2677 . . . . . . 7 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
21anbi1d 737 . . . . . 6 (𝑤 = 𝑧 → ((𝑥𝑤 ∧ (𝑥𝑧𝜑)) ↔ (𝑥𝑧 ∧ (𝑥𝑧𝜑))))
3 anabs5 847 . . . . . 6 ((𝑥𝑧 ∧ (𝑥𝑧𝜑)) ↔ (𝑥𝑧𝜑))
42, 3syl6bb 275 . . . . 5 (𝑤 = 𝑧 → ((𝑥𝑤 ∧ (𝑥𝑧𝜑)) ↔ (𝑥𝑧𝜑)))
54bibi2d 331 . . . 4 (𝑤 = 𝑧 → ((𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑))) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
65albidv 1836 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑))) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
76exbidv 1837 . 2 (𝑤 = 𝑧 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑))) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))))
8 ax-sep 4709 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑤 ∧ (𝑥𝑧𝜑)))
97, 8chvarv 2251 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606 This theorem is referenced by: (None)
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