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Theorem axsep2 3867
Description: A less restrictive version of the Separation Scheme ax-sep 3866, 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 3866 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 2098 . . . . . . 7 (w = z → (x wx z))
21anbi1d 438 . . . . . 6 (w = z → ((x w (x z φ)) ↔ (x z (x z φ))))
3 anabs5 507 . . . . . 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 1702 . . 3 (w = z → (x(x y ↔ (x w (x z φ))) ↔ x(x y ↔ (x z φ))))
76exbidv 1703 . 2 (w = z → (yx(x y ↔ (x w (x z φ))) ↔ yx(x y ↔ (x z φ))))
8 ax-sep 3866 . 2 yx(x y ↔ (x w (x z φ)))
97, 8chvarv 1809 1 yx(x y ↔ (x z φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1240  wex 1378
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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033
This theorem is referenced by: (None)
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