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Theorem axsep2 3846
 Description: A less restrictive version of the Separation Scheme ax-sep 3845, 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 3845 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(x y ↔ (x z φ))
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 2079 . . . . . . 7 (w = z → (x wx z))
21anbi1d 441 . . . . . 6 (w = z → ((x w (x z φ)) ↔ (x z (x z φ))))
3 anabs5 494 . . . . . 6 ((x z (x z φ)) ↔ (x z φ))
42, 3syl6bb 185 . . . . 5 (w = z → ((x w (x z φ)) ↔ (x z φ)))
54bibi2d 221 . . . 4 (w = z → ((x y ↔ (x w (x z φ))) ↔ (x y ↔ (x z φ))))
65albidv 1683 . . 3 (w = z → (x(x y ↔ (x w (x z φ))) ↔ x(x y ↔ (x z φ))))
76exbidv 1684 . 2 (w = z → (yx(x y ↔ (x w (x z φ))) ↔ yx(x y ↔ (x z φ))))
8 ax-sep 3845 . 2 yx(x y ↔ (x w (x z φ)))
97, 8chvarv 1790 1 yx(x y ↔ (x z φ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1224  ∃wex 1358 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-ext 2000  ax-sep 3845 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-cleq 2011  df-clel 2014 This theorem is referenced by: (None)
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