Theorem sbc3ie 3071

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
sbc3ie.1	|- A e. _V
sbc3ie.2	|- B e. _V
sbc3ie.3	|- C e. _V
sbc3ie.4	|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbc3ie	|- ( [. A / x ]. [. B / y ]. [. C / z ]. ph <-> ps )

Distinct variable groups:   x, y, z, A   y, B, z   z, C   ps, x, y, z

Allowed substitution hints:   ph( x, y, z)   B( x)   C( x, y)

Proof of Theorem sbc3ie

Step	Hyp	Ref	Expression
1		sbc3ie.1	. 2 |- A e. _V
2		sbc3ie.2	. 2 |- B e. _V
3		sbc3ie.3	. . . 4 |- C e. _V
4	3	a1i 9	. . 3 |- ( ( x = A /\ y = B ) -> C e. _V )
5		sbc3ie.4	. . . 4 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
6	5	3expa 1205	. . 3 |- ( ( ( x = A /\ y = B ) /\ z = C ) -> ( ph <-> ps ) )
7	4, 6	sbcied 3034	. 2 |- ( ( x = A /\ y = B ) -> ( [. C / z ]. ph <-> ps ) )
8	1, 2, 7	sbc2ie 3069	1 |- ( [. A / x ]. [. B / y ]. [. C / z ]. ph <-> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1372   e. wcel 2175   _Vcvv 2771   [.wsbc 2997

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-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sbc 2998

This theorem is referenced by: (None)