Theorem sbc3ie 3116

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 10	. . 3 |- ((x = A /\ y = B) -> C e. _V)
5		sbc3ie.4	. . . 4 |- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))
6	5	3expa 1151	. . 3 |- (((x = A /\ y = B) /\ z = C) -> (ph <-> ps))
7	4, 6	sbcied 3083	. 2 |- ((x = A /\ y = B) -> ( [.C / z ].ph <-> ps))
8	1, 2, 7	sbc2ie 3114	1 |- ( [.A / x ]. [.B / y ]. [.C / z ].ph <-> ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  _Vcvv 2860   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by: (None)