HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Theorem axsep2 2759
Description: A less restrictive version of the Separation Scheme axsep 2757, where variables x and z can both appear free in the wff ph, which can therefore be thought of as ph(x, z). This version was derived from the more restrictive ax-sep 2758 with no additional set theory axioms.
Assertion
Ref Expression
axsep2 |- E.yA.x(x e. y <-> (x e. z /\ ph))
Distinct variable groups:   x,y,z   ph,y

Proof of Theorem axsep2
StepHypRef Expression
1 a9e 1166 . 2 |- E.w w = z
2 ax-sep 2758 . . . 4 |- E.yA.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))
3 elequ2 1179 . . . . . . . . . . 11 |- (w = z -> (x e. w <-> x e. z))
43biimprd 161 . . . . . . . . . 10 |- (w = z -> (x e. z -> x e. w))
54pm4.71rd 650 . . . . . . . . 9 |- (w = z -> (x e. z <-> (x e. w /\ x e. z)))
65anbi1d 628 . . . . . . . 8 |- (w = z -> ((x e. z /\ ph) <-> ((x e. w /\ x e. z) /\ ph)))
7 anass 450 . . . . . . . 8 |- (((x e. w /\ x e. z) /\ ph) <-> (x e. w /\ (x e. z /\ ph)))
86, 7syl6bb 547 . . . . . . 7 |- (w = z -> ((x e. z /\ ph) <-> (x e. w /\ (x e. z /\ ph))))
98bibi2d 629 . . . . . 6 |- (w = z -> ((x e. y <-> (x e. z /\ ph)) <-> (x e. y <-> (x e. w /\ (x e. z /\ ph)))))
109albidv 1320 . . . . 5 |- (w = z -> (A.x(x e. y <-> (x e. z /\ ph)) <-> A.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))))
1110exbidv 1321 . . . 4 |- (w = z -> (E.yA.x(x e. y <-> (x e. z /\ ph)) <-> E.yA.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))))
122, 11mpbiri 201 . . 3 |- (w = z -> E.yA.x(x e. y <-> (x e. z /\ ph)))
131219.23aiv 1337 . 2 |- (E.w w = z -> E.yA.x(x e. y <-> (x e. z /\ ph)))
141, 13ax-mp 7 1 |- E.yA.x(x e. y <-> (x e. z /\ ph))
Colors of variables: wff set class
Syntax hints:   <-> wb 153   /\ wa 230  A.wal 995   = wceq 997   e. wcel 999  E.wex 1021
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-8 1005  ax-9 1006  ax-12 1009  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-sep 2758
This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022
Copyright terms: Public domain