Theorem sbc3ie 3106

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 1230	. . 3 |- ( ( ( x = A /\ y = B ) /\ z = C ) -> ( ph <-> ps ) )
7	4, 6	sbcied 3069	. 2 |- ( ( x = A /\ y = B ) -> ( [. C / z ]. ph <-> ps ) )
8	1, 2, 7	sbc2ie 3104	1 |- ( [. A / x ]. [. B / y ]. [. C / z ]. ph <-> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   _Vcvv 2803   [.wsbc 3032

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sbc 3033

This theorem is referenced by: (None)